;,'7/<? 


IN  MEMORIAM 
FLORIAN  CAJORI 


■r 


/»^e^ 


Text-Book  of  Algebra. 


THROUGH 


QUADRATIC  EQUATIONS. 


BY 


JOS.   V.    COLLINS,   Ph.D., 

rHOKKSS<»l!   OK   MATHKMATirs   IN   MlAMI    rMVKKSITV, 


CHICAGO: 

ALBERT,    SCOTT,   &  COMPANY. 

1893. 


By 
Albert,  Scott,  &  Company, 

CAJORl 


C.  J.  PETEB8  &  Son, 

Type-setteks  and  Electrotypees, 

Boston,  U.S.A. 


PREFACE 


Thk  usual  treatment  oi  algebra  blemls  irom  the  begin- 
ning the  two  distinct  conceptions  of  the  use  of  opposite 
numbei-s,  and  of  letters  to  stand  for  numbers.  In  the  fol- 
lowing pages  oi)i)osite  numbei-s  are  presented  first  in  the 
Ambic  notation,  all  the  rules  for  signs  being  given  before 
letters  are  introduced.  It  is  shown  further,  it  is  hoped  in 
a  simple  and  satisfactory  way,  how  the  operation  symbols, 
-j-  and  — .  can  be  used  to  distinguish  positive  and  negative 
numbers.  Careful  attention  is  i)aid  throughout  to  the  in- 
teri)retation  of  negative  and  imaginary  results.  Perhai)S 
the  most  important  feature  is  the  persistent  use  of  axioms 
in  the  solution  of  e([uations,  so  that  the  student  can  never 
forget  that  in  handling  an  e(piation  he  is  going  at  every 
step  through  a  process  of  reasoning.  This  course  secures 
t(j  the  student  greater  freedom  in  the  choice  of  methods  for 
the  solution  of  all  kinds  of  ecpiations,  and  2)repares  for  the 
examination  into  the  validity  of  processes.  Thus  the  treat- 
ment of  algebra  is  made  more  like  that  of  geometry. 

Throughout  the  whole  work  eveiy  subject  taken  up  is 
most  carefully  systematized.  The  demonstrations  in  nearly 
all  cases  are  rigorous,  and  not  mere  illustrations  of  what  is 
to  be  proved.  Since  algebra  is  one  branch  of  analysis,  one 
would  naturally  expect  that  its  demonstrations  and  exjila 
nations  would  be  given  in  analytic  form.  As  a  matter  of 
fact,  the  synthetic  method  is  often  largely  used  to  the 
student's  great  disadvantage.     If  the  subject  matter  be  first 


. Qi  i  2R1 


4  PREFACE. 

carefully  arranged,  and  the  demonstrations  then  put  in  the 
analytic  form,  the  author  believes  a  good  presentation  is 
attained.  The  different  parts  of  the  book  are  connected  as 
closely  as  x)ossible  by  cross  references,  so  that  each  is  made 
to  throw  light  on  the  others.  Aside  from  the  exceptions 
noted,  the  book  does  not  differ  much  in  arrangement  and 
method  from  other  works  of  a  similar  scope  in  current  use. 

As  the  algebraic  notation  is  made  use  of  in  all  or  nearly 
all  of  subsequent  mathematical  study,  it  is  of  great  impor- 
tance that  the  student  should  be  thoroughly  acquainted  with 
its  details  and  experienced  in  its  use.  The  teacher  should 
endeavor  to  make  the  student  as  familiar  with  it  as  possible. 
The  interest  awakened  in  the  study  of  equations  usually 
insures  success  in  that  part  of  the  subject.  Enough  material 
is  included  here  to  constitute  a  good  high  school  course,  as 
also  to  fit  for  the  best  colleges. 

Excepting  a  considerable  number  which  are  original, 
the  exercises  have  been  drawn  from  a  great  variety  of 
sources,  and  are  written  in  all  legitimate  notations.  It 
would  be  very  difficult  to  give  proper  credit  for  them. 
Special  acknowledgment,  perhaps,  ought  to  be  made  for 
those  taken  from  Heis's  collection.  The  author  is  greatly 
indebted  to  his  former  instructor  and  esteemed  friend.  Pro- 
fessor S.  J.  KiRKwooD,  LL.  D.,  of  the  University  of  Wooster, 
for  valuable  counsel  and  interest  shown  during  the  whole 
time  of  preparation.  References  to  aid  received  from  other 
sources  are  made  in  the  text. 

April,  1893. 


SUGGESTIONS  TO  TEACHERS. 


Ix  addition  to  exercises  in  the  text  others  may  be  assigned 
in  drill  books  such  as  Perrin's  or  Ray's,  or  on  algebra 
tablets.  The  latter  are  convenient  for  use,  and  are  not 
very  expensive.  The  rigorous  classification  of  the  subject 
matter  has  thrown  some  demonstrations  and  exercises  in 
places  where  they  should  be  used  only  on  the  second  read- 
ing. It  is  thought  that  more  is  gained  to  the  student  in 
clearness  of  conception  of  the  subject  than  is  lost  by  the 
extra  difficulty  he  finds  in  mastering  such  parts.  Material 
which  may  be  omitted  is  marked  with  a  star.  That  which 
ouglit  to  be  omitted  by  young  students  with  a  double  star. 
The  teacher  is  the  best  judge  of  what  any  particular  class 
can  do,  and  if  any  of  the  exercises  are  clearly  too  hard 
they  should  be  passed  over.  With  a  very  few  exceptions 
the  exercises  will  be  found  to  be  no  more  difficult  than 
those  of  the  best  books  in  common  use. 

Not  enough  of  the  history  of  the  different  branches  of 
mathematics  has  been  taught  in  American  schools.  Much 
interest  can  be  added  to  recitations  now  and  then  by  brief 
accounts  of  the  historical  development  of  subjects  under 
consideration.  The  author  had  thought  of  giving  a  short 
resume  of  the  history  of  algebra,  but  afterwards  became 
convinced  that  it  would  be  preferable  to  ask  teachers  them- 
selves to  go  over  this  history  pretty  thoroughly,  and  then 
introduce  the  results  of  their  reading  in  the  class-room 
as  occasion    offered.     For  such   preparation   Ball's  "  Short 

5 


6  SIGGESTIONS    TO    TEACHERS. 

History  of  Mathematics"  (Macmillan  &  Co.,  New  York 
City)  is  the  best  book  attainable,  and  it  will  be  found  to 
contain  a  list  of  other  works  on  the  subject  should  any  one 
desire  to  prosecute  the  study  further.  Chrystal's  Text-book 
contains  many  historical  note?,  and  is  besides  one  of  the 
fullest  and  best  treatises  on  algebra  in  the  English  lan- 
guage. In  addition  to  such  historical  inquiries  a  study  of 
methods  of  teaching  algebra  is  also  advised.  For  this 
purpose,  Dr.  Fr.  Reidt's  '^Anleitung  zum  mathematischen 
Unterricht  an  hoeheren  Schulen  "  (Berlin :  Grote)  is  recom- 
mended. Circular  of  Information,  No.  3,  1890,  of  the 
Bureau  of  Education,  prepared  by  Florian  Cajori,  is  on  the 
teaching  and  history  of  mathematics  in  the  United  States. 


TABLE    OF    CONTENTS. 

{^YyOPiylS  FOR  RE  VIEW  a.) 


INTRODUCTION. 
CHAPTER  I.  —  Algebra  as  a  Branch  of  Mathematics. 

AUTICI-K.  I'AOE. 

1.  Definition  of  Mathematics 2.5 

a.  Word  Quantity 25 

2.  Definition  of  Geometry 26 

3.  Definition  of  Arithmetic 26 

a.  Illustration  of  Difference  between  Algebra  and  Arithmetic,  26 

4.  Definition  of  Algebra 26 

5.  Letters  in  Algebra 2(J 

6.  Two  Kinds  of  Numbers 26 

7.  Plan  of  Presentation 27 

FIRST  GENERAL  SUBJECT  — THE  ALGEBRAIC 
NOTATION. 

CHAPTER  IL  — On  Opposite  Numbers. 

8.  Numbers  in  Algebra  separated  into  two  Classes 28 

Section  I.  —  Nature  of  Opposite  Numbers.  —  Opposite  Numbers 
and  the  Signs  of  Addition  and  Subtraction. 

9.  Failure  in  Arithmetical  Analysis 28 

10.  The  Arithmetical  Infinite  Series 29 

11.  An  Opposite  Series 29 

12.  The  Series  Used 29 

13.  Marking  the  two  Kinds  of  Numbers 29 

14.  Instances  of  Opposite  Numbers IJO 

I').  Definitions  of  Addition  and  Subtraction  in  Arithmetic     .     .  30 

16.  Algebraic  Addition 30 

17.  Algebraic  Subtraction 31 

a.  Distance  on  Scale 31 

18.  Exercise  in  Addition  and  Subtraction 32 

19.  Meaning  attached  to  Zero  in  Algebra 32 

20.  The  Use  of  other  Signs  to  mark  the  Series 32 

21.  The  0  in  the  Series  Sign  is  dropped 33 

22.  Conclusion 33 

2:>.  Names  of  the  Series 33 

7 


TABLE    OF    CONTEXTS. 


24.  Two  Views  of  the  Signs  +  and  — 38 

25.  Positive  and  Negative  Numbers.     Plus  and  Minus  ....     34 

26.  Like  and  Unlike  Signs 34 

Section  II.  —  The  Fundamental  Rules  Extended  to  apply  to 
Opposite  Numbers. 

27.  Rules  for  the  Fundamental  Operations  applicable  to  Opposite 

Numbers 34 

.  .  I.  —  ADDITION. 

28-30.  Addition  of  Positive  and  Negative  Numbers.     Examples, 

Rule,  Exercise     .     .     .     .  • 34 

II. — SUBTRACTION. 

31-33.  Subtraction  of  Positive  and  Negative  Numbers.     Exam- 
ples, Rule,  Exercise • 38,  39 

III.  —  MULTIPLICATION. 

34.  Def.  of  Multiplication.     Illustration,     a.  Explanation  ...     40 

35.  Investigation  of  the  Reasons  for  the  Rule  of  Signs  in  Multi- 

plication.    Two  Cases 41 

36-37.  Rule  and  Exercises  in  the  Multiplication  of  two  Factors,     42 

38.  The  Commutative  (two  cases)  and  Associative  Laws  in  Mul- 

tiplication    42 

39.  The  Sign  of  the  Product  when  three  or  more  Factors  are 

multiplied  together 43 

,    .1.  The  Product  of  any  Number  of  Positive  Factors ....  43 

2.  The  Product  of  an  Even  Number  of  Negative  Factors      .  43 

3.  The  Product  of  an  Odd  Number  of  Negative  Factors    .     .  44 
40-41.  Rule  and  Exercises  in  the  Multiplication  of  more  than 

two  Factors 44 

42.  Powers  of  Algebraic  Numbers.     Definition 46 

a.  Index  of  Power 46 

1.  Rule  for  Signs.    2.  Exercise 46,47 

IV. — DIVISION. 

43-44.  Definition,  Rule  of  Signs,  Exercise 47,  48 

45.  Roots  of  Opposite  Numbers 48 

a.  Indices  and  Radical  Signs .     .     »     ; 48 

b.  The  Double  Sign  ± 49 

1.  Square  Root 

(1)  of  +  Number  \  .^ 

(2)  of  -  Number  (.:::: ^* 

Cube  Root  •  ... 

(1)  of  +  Number  \.     .     .  .^ 

(2)  of  -  Number  ( *^ 

3.  Other  Roots 

j  (1)  Index  a  Prime  Number        }  .q 

I  (2)  Index  a  Composite  Number  j 

4.  Rules  for  Signs  of  Roots,  (1.  (2.  (3 49 

46.  Exercise  in  Extracting  Roots 50 


TAI5M-:   OF   CONTKNTS. 


(HAI'lKi:     III.  —  LkIIKUS    used   to    SniNlFY    NlMBEUS.— 

Definitions  and  Explanations. 

AKTUI.i;.  I'AiiK. 

47.  Use  of  Letters  the  Second  Distinguishing  Characteristic  of 

Algehra.     Problems  to  illustrate  this 51-53 

Section  I.  — Letters  loith  the  Signs. 

48.  Definition  of  Algebraic  Expression.     Illustrations   ....  .");} 

49.  List  of  Signs  used  in  Arithmetic  and  Algebra .');{ 

50.  The  Sign +.    Illustration :>4 

51.  The  Sign  — .'>4 

52.  The  Sign  X 54 

a.  The  Period  as  a  Sign 54 

b.  No  Mark.     Explanation 54 

bH.  The  Sign  -f.     a.    F'ractional  Fonn 55 

54.  The  Parentheses 55 

55.  The  Equality  Sign 55 

a.  Explanation.     Example 55 

50.  An  Exponent .56 

a.  No  Exponent  written,  1  understood 56 

b.  Meaning  and  Treatment  of  Fractional  Exponents      .     .  56 

57.  Power  of  a  Number.    Definition    . 5(J 

a.  An  Exponent  is  called  the  Index  of  a  Power  ....  56 

b.  Second  Power  is  called  the  Square ;  third  Power  is  called 
the  Cube 5() 

58.  The  Radical  Sign,  y/ 5<} 

59.  Index  of  Radical 56 

a.  No  Index  written,  2  understood 56 

60.  Definition  of  lioot 57 

a.  Square  Root  and  Cube  Root 57 

Section  II.  —  Classification  of  Symbols. 

61.  Definition  of -Si/m6o/.s.    a.  Kin<ls 57 

62.  Symbols  of  Number 

(  Numbers  ) 

h     i " 

as.  Of  Operation 57 

64.  Of  Relation 
of  Inequality  ) 

of  Equality     [ .58 

of  Variation    ) 

65.  Of  Aggregation.     List ,58 

66.  Of  Omission 58 

67.  Logical  Symbols,  .*.  and  •.• .58 

Section  III.  —  Classyication  of  Algebraic  Expressions. 

68.  Term,     a.  Simple,     6.  Compound 50 

69.  Monomial,     a.  Term  and  Monomial 5!> 

70-72.  Binomial,  Trinomial,  Polynomial 59 


! 


10  TABLE   OF   CONTENTS. 

ARTICLE.  Section  IV.  —  On  the  Term.  page. 

73.  Coefficient  and  Literal  Parts  of  a  Term 60 

74.  The  Coefficient 60 

a.  Coefficient  and  Exponent  contrasted 60 

b.  Tlie  Coefficient  1  not  written 60 

75.  The  Literal  Part 60 

a.  Dimensions  of  a  Term 60 

76.  Similar  Terms 61 

a.  Addition  and  Subtraction  of  Similar  Terms      ....  61 

b.  Like  and  Unlike  Signs  of  Terms 61 

77.  The  Degree  of  a  Term 61 

a.  The  Exponent  1  understood  must  be  counted   ....  61 

78.  Homogeneous  Terms 61 

a.  A  Polynomial  Homogeneous 61 

CHAPTER  lY.  —  Exercises  in  Notation. 
Section  L  —  Exercise  in  Beading  Algebraical  Expressions. 

70.  Exercise 62 

Skction  it.  —  Exercise  in    Writing  Algebraical  Expressions. 

80.  Exercise .  63 

Section  III. — Numerical  Values. 

81-82.  Definition  of  Numerical  Values.     Exercise 64 

83.  Numerical  Values  of  Similar  Terms.     Exercise 64 

84.  Numerical  Values  used  to  verify  Equations 6.5 

8.").    Vo  find  the  Numerical  Value  of  a  Letter  in  an  Equation  .     .  65 

CHAPTER    v.  — Addition. 

8(;.  Definition  of  Addition 69 

a.  Meaning (59 

87.  Examples  of  Addition 61) 

1.  Terms  Similar  and  with  Like  Signs  (l.-(2 69 

2.  Terms  Similar  with  Unlike  Signs  (1. -(2 70 

3.  Not  Similar,  or  Mixed 71 

4.  Wlieii  Terms  to  be  added  constitute  a  Single  J^olynomial  .  71 
88-89.  Uule  (1-5).     Exercise "..■....  72 

CHAPTER   VI.  —  Sir.THACTiON. 
J)0-93.  Definition,  Examples  (1,  2),  Rule  (1,  2),  Exercise      .     .  74-7() 

CHAPTER   VII. — Symuoi.s  of  Acjoheoation.     Ar.so 

Exercises  in  Addition  and  Si utijaction. 

114.  Symbols  of  Aggregation.     Definition 77 

95.  (^lantity  in  Algebra.     Definition 77 

a.  Quantity,  Algebraic  Expression,  and  Number  ....  77 

Section  I. — "On  Compound  Terms. 

96.  Addition  and  Subtraction  of  Similar  Com])ound  Terms  with 

Numerical  Coefficients 78 


TAHLK    <»K    «<>N  PKNTS.  1  1 

ARTICLK.  I'AtiK. 

J)7.  Formation  of  rompound   rooflieients  for  Common  Literal 

Part 7S 

OS.   Ff)nnati(m  of  Conipoiiiiil  ('ootti<-i«Mits  for  Similar  ('onj])oiiiiil 

IVruis 7'' 

Skc'TIon  II.  —  Principles  and  liulis  votinectetl  wif/i  the  I'sc 

of  the  St/tnhol.s  of  A<jgr€(i<ifioh. 

09.  Explanation  of  Tse  of  Symbols 70 

1.  To  indicato  Addition  and  Subtraction  of  Polynomials.  70 

2.  To  enclose  Series  Sijjrn SO 

100.  Kules  for  Synd)ols  of  Aggregation.    1-4.    Note SI 

Skction  III.  —  Exercise  in  Jiemoving  Si/inhnl.'<. 

101.  IJenjoving  Symbols  from  SiniX)le  Expression SI 

102.  Kemovinjx  Symbols  from  ('omi)lex  Expression sj 

Se(;ti<>x  IV. —  r^es  of  the  S//nihnlfi. 

10:5.  General  Exercise S:J 

C'HAPTElt     Vlli.  Ml    I.TIIM.K   ATIMN. 

104.  Definition     a.  Literal  Proof  of  Kiilc  for  Si«:ns S.") 

105.  Rule  for  Signs SO 

KKi.  Multii)lication  of  Monomials S(» 

1.  The  Signs  (1. -(4 S<» 

2.  Numeric'al  Cot'tticient SO 

:^,.  The  Literal  Part,  (1.(2.  (', so.  ST 

4.  Examples ST 

107-108.   Rule,  1-:*..     Exercise SS 

101>.   Multiplication  of  Polynomials SO 

1.  Multiplication  of  Sum  by  a  Single  Ttiiii SO 

2.  Multiplication  of  Tenu  by  a  Sum SO 

:).   Multiplication  of  two  Sums SO 

4.   Mtdtiplication  of  m — »i  by  / — )/ 00 

.'>.   Distributive  Law 00 

110-112.    Examples,      ifule.     (Monomials,   Polynomial    ;in  I    .Mo- 
nomial, Polynomials  (1  and  (2.     Exen-ise IM)  0:i 

11:1.   Tse  of  one  L«'tter  to  stand  for  a  (Quantity 04 

rt.  A  Fonnula 0.') 

114.  Three  Theorems  in  Multiplication.    Proofs  and  Examples.  0."»  07 

11').   Exerci.se  in  the  I7se  of  the  Theorems 07 

lie*.  Other  Noteworthy  Examples  in  .Midlipli'-.itio '.»S 

1.  S(juare  of  a  Trinomial  Sum OS 

2.  Sijuare  of  a  Polyiunuiai OS 

'•).  Tlieon'm  W OS 

4.  Cube  of  a  Hinondal OS 

r>.  Theorem  V OS 

(J.   Fourtli  Power  of  a  Pinomial OS 

7.  Simple  Multiplications 00 

8.  Tln'onnn  VI 00 

0.  Exercise 00 


12  TABLE   OF   CONTENTS. 

ARTICLE.  PAGE. 

117.  Simple  Powers.     Definition 09 

1.  Monomials,  Examples,  Rule,  Exercises 09 

2.  Polynomials 100 

(1.  Square  of  Binomials,  Trinomials,  Polynomials  .     .     .  100 
(2.  Cubes,  Fourth  Powers,  etc.    References 101 

CHAPTER  IX.  —  Divisiox. 

118.  Definition 102 

119.  Monomials,    a.  Two  Ways  of  Writing 102 

1.  The  Signs,  Reason,  Rule 102 

2.  The  Coefficient 102 

3.  The  Literal  Part.     Particular  Cases 102 

(1.  Letter  in  Dividend  and  not  in  Divisor 102 

(2.  Letter  having  same  Exponent  in  both  Terms ....  102 

(3.  Letter  with  Greater  Exponent  in  Divisor 103 

4.  Examples 103 

120-121.  Rule  for  Monomials.     Exercise 103-104 

122.  Leading  Letter  in  Polynomial  ......     ."^    ...     .  104 

123.  A  Polynomial  arranged 10.5 

124.  Division  of  Polynomials 10.5 

1.  Polynomial  by  Monomial.     Example  and  Explanation    .  105 

2.  Polynomial  by  Polynomial.     Example  and  Explanation,    105 

a.  Nature  of  Long  Division  in  Algebra 106 

b.  Necessity  of  Dividing  First  Term  of  Dividend  by  First 
Term  of  Divisor,  etc. 107 

125-127.  Examples,  a,  b,  c,  d.     Rule,  1,  2,  3.     Exercise  (1,  (2, 

(3,  (4, 107-109 

128.  Zero  and  Negative  Exponents.     Theorems  of  the  Notation,  110 
1.  Zero  Exponents.  2.  Negative  Exponents.  3.  Exercises,  111-113 

129.  Simple  Roots.     Definition  (45) 113 

1.  Monomials,  (l-(3.  Example,  Rule,  Exercises.  (4.  Dif- 
ferent Ways  of  Regarding  the  Same  Power 114 

2.  Polynomials.     (See  Chap,  xviii.) 114 

130.  Four  Theorems   in    Division.      Divisibility  of    Binomials, 

1-4 115-116 

5.  Examples  in  the  Use  of  the  Theorems  of  Division,  with 
Particular  Reference  to  Powers  of  Quantities  which  are 
themselves  Powers 117 

131.  Exercise  on  the  Theorems  of  Divisibility 118 

CHAPTER  X.— Factoring. 

132.  Definitions  of  Terms  used 120 

1.  Divisor.  2.  Multiple.  3.  Prime  Quantity.  4.  Com- 
posite Quantity 120 

1.33.  Monomials,  (1.     Standing  alone,  (2 120 

134.  Binomials 121 

1.  Difference  of  two  Squares 121 

2.  Difference  of  the  same  Powers 122 

3.  Difference  of  the  same  Even  Powers 122 

4.  Sum  of  the  same  Odd  Powers 123 

5.  Binomials  separable  into  two  Trinomial  Factors     .     .     .  123 


TABLK   OF   C(>M1:NT.S.  ^  1  H 

ARTK  I.K.  1A<.I;. 

135.  Trinomials 124 

1.  TriiiouiiHl  Squares 124 

2.  Siiiiph'  Trimmiials  not  Sciuart's 125 

;5.  Trinomials  th«»  Product  of  any  two  Binomials.     1,  2,  :J, 

Examples,  Kule,  Exercise 127-128 

4.  Trinomials  of  the  Form  A^+aA^B^+B^ 12« 

5.  Trinomials  the  Proiluct  of  a  Binomial  and  Trinomial     .  128 

136.  Quadrinomials 121) 

1.  The  Cube  of  a  Binomial.  Explanation,  Rule,  Exerciser,  121)-130 

2.  Quadrinonnals  the  Difference  of  two  tSiiuares     ....  130 

3.  Quadrinomials  the  l*roduct  of  two  Binomials      ....  131 

4.  Quadrinomials  the  Product  of  BiiHjmials  and  Trinoimals,  llHi 

137.  Polynomials  of  more  than  Four  Terms      .     » 134 

1.  Expressions  which  are  Perfect  Powers.     Note   .     .     .     ,1:54 

2.  Polynomials  the  Difference  of  two  Squares 1:54 

3.  Polynomials  the  Square  of  a  Trinomial I:i5 

4.  Other  Polynomials IHT) 

General  Remark.  —  This  Classification  requires  that  all  expressions 

must  be  considered  from  the  standpoint  of  their  Developed  Forms. 

138.  Promiscuous  Exercise  in  Factoring 130 

CHAPTER  XI.  —  Common  Factoks. 

139.  Common  Factors,  II.  C.  F 138 

1.  Common  Factors,  Definition 138 

2.  Highest  C'ommon  Factor 138 

a.  g.  c.  d.  and  h.  c.  f 138 

Skction  i.  —  First  Method.  —  The  h.  c.  f.  by  factoring. 
140-142.  Principles  involved  in  finding  the  h.  c.  f.  by  factoring. 

Rule  (1,  2,  3),  Exercise,  Note,  Miien  adapted  .     .     .      138-140 

Section  II.  —  Second  Method.  — By  Continued  Division. 

143.  Principles   involved   in  finding  the  h.  c.  f.  by  Continued 

Division 141 

1.  Principle  of  First  Metluxl 141 

2.  A  Divisor  of  a  Quantity  is  a  Divisor  of  any  Multiple  of  it,  141 

3.  A  Common  Divisor  of  two  Numbers  is  a  Divisor  of  their 
Sum  or  Difference 141 

144.  Application  of  Principles  to  justify  Metho<l 142 

1.  Arithmetical  Example 142 

2.  General  Demonstration 14:i 

145.  Method  and  Principles  in  the  Solution  of  Algebraical  Prob- 

lems.    Modifications,  1,  2,  3 144 

140-148.  Rule.     1-4,  a  — d.     Examples.     Exercise     .     .      147-149 

CHAPTER  XII. —  Common  Multiples. 

149.  Common  Multiples.     The  1.  c.  m.     1.    A  Multiple.     2.  A 

Common  Multiple.    3.  The  1.  c.  m.    a.  Remark,     b.  Uses,  151 

1.50.  Principles  applicable  in  finding  Common  Multiples.  1.  Mul- 
tiples.    2.  The  Product.     3.  The  1.  c.  m •   .     .  152 


14  TABLE    OF   C0NTP:NTS. 

articij:.  pack. 

151-15H.  Kiiles.  1.  By  factoring  the  Quantities.  2.  Special 
Process.  :5.  When  the  Quantities  cannot  be  factored.  (1. 
When  there  are  but  two  Quantities.  (2.  When  tliere  are 
more.     a.  b.  Remarks.    Examples,  Exercise    .     .     .      152-1.54 

•       CHAPTER   XIII.  —  rKACTio>.s. 

154.  Definition,     a.  Illustration,     h.  An  Indicated  Division  .     .  156 

SECTION  I.  —  Classification  and  Principles. 

155.  Classification  of  Fractions 157 

1.  Witli  respect  to  their  Origin.     (l.-(o.     Simple,  Complex, 
Mixed  Quantity 157 

2.  With  respect  to  their  Capability  of  Reduction  to  a  Mixed 
Quantity.     (1.  Proper.     (2.  Improper 157 

156.  Fundamental  Principle  in  Fractions 158 

157.  The  Three  Signs 158 

1.  Effect  upon  the  Fraction  of  changing  these  Signs.     (l.-(5. 
Cases 159 

2.  Rules.     (1.  (2 159 

Sectiox  II.  — deduction. 

158.  Definition.     Kinds 100 

I. — TO    LOWKST    TKHMS. 

159-160.  Reduction  to  Lowest  Terms,     Explanation  and  Exercise  160 

II. — TO    MIXED    OK    EXTIKK    QlAXTrrV. 

1()1-162.  Reduction  to  Mixed  or  Entire  Quantity  ....     161,  162 

16:^-164.  Dissection  of  Fractions 162 

165-166.  Fractions  written  as  Entire  Quantities 163 

III. — MIXED    Ql'AXTITIES    TO    nil'HorEIi    FHA(   I'lONS. 

167-168.  Reduction  of  Mixed  Quantities  to  Iiiipiojx'r  Frac- 
tions  168,  164 

IV. —  i;Ki)r<  Tiox   or    fi!A(ti()Xs    ojt   ixteoeks   to   kqi'iva- 

I.KXT    1  JLVCTIOXS    IlAVIXCx    (HVEX    DKXOMIXATOIiS. 

169-170.  To  reduce  a  given  Fraction  or  Integer  to  an  Equiva- 
lent Fraction  having  a  given  Denominator      165 

171-172.  To  reduce  two  or  more  Fractions  to  Equivalent  Frac- 
tions having  a  Common  Denominator.  L'^sually  the 
1.  c.  d 166,  167 

Sectiox  hi.  —  The  Fundaxicntal  lialcs  in  Fractions. 
Ho.  Definition.     Explanation 168 

I. ADDITIOX. 

174-175.  Addition,  Rule,  Exercise Ji'.s,   KiO 

ir.  SIIJ'il{A<  TIOX. 

17<5-177.   Subtraction,  Rule,  Exercise 170,  171 

111.  —  Ml  I/!1IM.[(  A  riox. 
178-179.   Multiplication,  Rule.  ^^  />.  Ivxercise 173 


TABLE   OF  CONTENTS. 


ARTin.K.  IV.  —  i)lVISIi>N.  l'A«iK. 

180-181.  Division,  Kule,  Exercise 17'. 

182-18:3.  Complex  Fractions,  Definition,  ExjUanation,  Exercise  .  17<> 
184.  Promiscuous  Exercise ITS 

SECOND   GENERAL   SUBJECT.  —  SIMPLE 

EQUATIONS. 

CHAPTER  XIV.  — SiMPLK  Equations  avitii  Onk  ^^K^<>^v^. 

Skctiox  I.  —  General  Definitions. 

lHr>.  Definition  of  an  Equation IM* 

18(J.  Identical  and  Conditional  Equations 1S2 

187.  Identical  Equations 1S2 

a.  Character,    h.  Identity  Sign,  rrr 18:J 

188.  Conditional  Equations 188 

180.  Explanation  of  Conditional  Equations 183 

a.  Verifying,     h.  Simplest  Case.     c.  Importance  of  the 

Equation  in  Algebra 188,  184 

190.  Members  of  an  Equation 1S4 

191-15)2.  Numerical  and  Literal  Equations 184 

198.  To  Solve  an  Equation 184 

194.  Known  and  I^nknown  Quantities  in  Equations 185 

195.  The  Degree  of  an  Equation 185 

190.  Simple  Equations,     a.  Remark 185 

Section  II.  — Algebraical  Method  of  Treatment. 
197.  Nature  of  the  Treatment 185 

I. — LOGICAL   TKKM.S. 

198-200.  Proposition,  Problem,  Theorem,  I)«M)i«>n^f?;tti<)ii.  (  or<>l- 

lary,  Scholium,  Hyjwthesis,  Axiom Ix". 

II. — AXIOMS. 

207.  Axioms 18() 

a.  Addition  and  Subtraction 180 

1.  Equals  added  to,  etc 180 

2.  Subtraction 18<> 

6.  Multiplication  and  Division 180 

8.  Multiplication.     4.  Division 1S(> 

c.  Powers  and  Koots 187 

6.  Powers.     0.  Kwts 187 

7.  Quantities  equal  to  the  same  Quantity,  etc     .     .     .187 

8.  General  Axiom 1ST 

III.  —  SOLITION    OF    EQUATIONS. 

20^209.  Solution  of  the  Ecjuation,  ax  =  b.  Exercise  .     .     .     187,   188 
210-211.  Solution   of   Integral   Equations.     Theorem  of  Transr 

position.    Exercise 188-190 

212-218.  Solution  of  Fractional  Equations.    Theorem.     Clearing 

of  Fractions 191,  192 

( 'on.  I.   Signs  of  all  Terms  may  be  changed 191 

SciioL.  I.    Any  Common  Multiple  may  be  used     .          .     .101 
ScHOL.  II.    General  Kemark.     Examples 191 


16  TABLE    OF    CONTENTS. 

AKTICLP:.  I'AGK. 

214.  Examples  of  Complete  Solution UK] 

21.5.  General  Rules  for  Solving  Simple  Equations 104 

1.  Normal  Process,  (l-(4 11)4 

2.  General  Process,  (1  (2.  llemark 11)4 

210.  Exercise 104 

217.  General  Kemarks  on  Solution  of  Equations   ...     .     .     .  100 

Section  111. — Froblenis. —  One  Unknown. 

218.  Explanation  of  Problems  of  Algebra.     Analytic  JVletliod      .  200 

219.  Solution  of  Problems.     Two  Operations 200 

1.  Statement.  2.  Solution 200,  201 

220.  In  the  Enunciation  we  may  see 201 

1.  Description  of  an  Unknown 201 

2.  Assertion  of  Equality 201 

221.  Function  of  a  Quantity 201 

222.  General  Rule  for  solving  Problems 201 

1-5.     a,  6 201,  202 

223.  Proportions  turned  into  Equations 202 

224.  Exercise 202 

CHAPTER  XV.  —  Simple  Equations  Containin(;  Two 
oil  Moke  Unknowns. 

225.  Simultaneous  Equations 213 

a.  Conditional  Problems  solved  with  one  or  with  more 
Unknowns 213 

b.  Indeterminate  Equations  . 214 

c.  Any  Proper  Axiomatic  Operation  does  not  change  the 
Identity  of  an  Equation 214 

226.  Elimination,     a.   Kinds 215 

Section  I.  —  Simple  Equations  with  Two  Unknowns. 

227.  Method  of  Solution 215 

I. — SUBSTITUTION. 

228.  Elimination  by  Substitution 215 

229.  Substitution  to  find  Value  of  other  u^nknown 215 

230.  Exercise 216 

II.  —  COMPAKISGN. 

231-232.  Elimination  by  Comparison.     Exercise     ....    218,  219 

III. — ADDITION    AND    SUIiTH ACTION. 

233.  Elimination  by  Addition  and  Subtraction.    Definition.    Ex- 

planation   220 

1.  To  make  Coefficient  the  same  (l-(3 220 

2.  Adding  or  Subtracting 221 

234.  •  Exercise 221 

IV.  —  SPECIAL   METHODS    OF   ELIMINATION. 

235.  Special  Methods 224 

A.  Elimination  by  Undetermined  Multipliers 224 


TAliLK    OF    CoNTKNTS.  17 

AKTK  I.K.  I'A«ii;. 

•_';'.»;.  <  it'iieral  Process  by   Uiuletenuiued  Multipliers.     Bezout's 

M«*thc)(l.     Exercise 224 

IJ.  Eliinliiation  by  means  of  one  or  more  Derived  E<iuations,  225 

237.  Explanation  of  Elimination  by  Derived  Eijuations.     Exer- 

cise   225,  2:5(5 

('.  Elimination  by  lindinj;  tbe  h.  c.  f 22<> 

238.  Methoil  and  Principle  by  finding  tlie  li.  c.  f 22(5 

231).  (ieneral  Exercise 227 

240.  Problems 22i) 

Section  II. — Shnultaneous  Equations  contniniiKj  Three  or 
more  Unknowns. 

241.  Metliod  of  Solution.    Choice  of  Plan 2;W 

242-245.  Examples,  Kule  1,  2,  3,  a,  6,  Exercise,  Problems  .    233,  238 

CHAPTER    XVI.  —  Indeterminate    and    Redundant    E<iUA- 

TION8,  Nb:gative  Solutions,  Generalization. 

Section  I.  —  Indeterminate  Equations. 

24(5-247.  Definition,  Examples 24() 

248-24S>.   Kule  1-4.     Exercise 242,  243 

250.  Problems.     Diophantine  Analysis 244 

Skctiox    II. — Redundant  Equaiions. 

251.  Definition.  Two  Kinds 245 

1.  Compatible.      2.   Incompatible,     a.  Reference  ....  245 

Section  HI. — Negative  and  Inconsistent  Solutiojis  in  Simple 
Equations. 

252.  Problems  involving;  Arithmetical  Inconsistencies    ....  245 
2.53.  Lessons  from  the  Examples  of  the  Preceding  Article  .     .     .  240 

1.  Significance  of  Negative  Results 249 

2.  Positive  Results  are  Impossible  in  some  Problems,  due  to 
the  Presence  of  Negatives  in  some  Stage  of  the  Solution  .  24t) 

Section  IV.  —  Literal  Problems.  —  Generalization. 

254.  Definition,    a.  Explanation 24l> 

255.  Exercise,     a,  b.  Explanations 241> 

THIRD  GENERAL  SUBJECT.  — NOTATION  CON- 
CLUDED.—POWERS,  ROOTS,  AND  RADICALS. 

CIIAPTEK    XVII.— Of  Powkrs. 

25(5.   Involution 25(> 

Section  I.  —  Monomial  Poivcrs. 

257.  Exercise,  Reference,     a.  Fractions 250 

Section  II.  —  Binomial  Powers. 

258.  Investigation   of   Binomial   Powers.      Newton's   Theorem, 

1-5 257,  2(50 

a.  Meinoriter  Rule.      b.  Remark 2(J0 

259-200.  Examples  of  use  of  Binomial  Theorem.     Exercise    .     .  200 


18  TABLE   OF   CONTENTS. 

ARTICLE.  Section  III.  —  Polynomial  Powers.  t'aok. 

261.  Development  of  Polynomial  Powers      .     .     r 262 

a.  Polynomial  Squares 262 

CHAPTER  XVIII.  —  Of  Exact  Roots. 

262.  Evolution,  Definition,     a.  Division  of  Chapter 265 

S PACTION  I. — Monomial  Boots. 

263.  Roots  of  Monomials 26.5 

264.  Signs  of  the  Roots.     Two  Square  Roots 265 

265.  Imaginary  Quantities 265 

266.  Exercise  in  Extracting  Real  Roots  of  Monomials    ....  266 

Section  II.  —  Square  Root  of  Polynomials. 

267.  Square  Root  of  Polynomials.     How  effected,     a.  Explana- 

tory    266 

268-269.  Rule  1-3.    Exercise 268 

270.  Extraction  of  the  Square  Root  of  Arithmetical  Numbers      .  269 

271-272.  Rule  1,  2,  3.     a,  6,  c.    Exercise 271,  272 

Section  III.  —  Cube  Hoot  of  Polynomials. 

273.  Extraction  of  Cube  Root  of  Polynomials,  1,2 272 

274-275.  Rule  1,  2,  3.     Exercise 273,  274 

276-278.  Extraction  of  the  Cube  Root  of  Arithmetical  Numbers, 

1,  2.    Rule  1-5.     Exercise 274,  276 

Section  TV.  —  Extraction  of  other  Boots  of  Polynomials. 

279.  Other  Roots  by  Square  and  Cube  Root  Processes    ....  276 
1.  Fourth  Root.   2.  Sixth.     3.  Eighth.      4.   Ninth,  etc.    .     .  276 

280.  Prime  Roots  other  than  the  Square  and  C'ube  Roots.    The 

Fifth  Root  as  an  Example    .     .     .     . 277 

281.  Exercise 277 

CHAPTER   XIX.  ^ — Of  FifACTioNAi.  Exponents. 

282.  Origin,     a.  Illustration 278 

283.  Meaning  of  Terms 278 

284.  Fundamental  Principle  Governing  use.  —  Root  of  Product 

=  Product  of  Roots 278 

a.  Order  Indifferent 279 

b.  Other  Terms  of  Exponent 279 

285.  Fractional  Exponents  and  Radical  Sign 279 

286.  Object  of  Separate  Treatment 280 

1.  To  show  same  Rules  apply 280 

2.  As  an  Exercise  in  their  use 280' 

287.  Fractional  Exponents  and  the  Fundamental  Rules      .     .     .  280 

1.  Addition  and  Subtraction 280 

2.  Multiplication  and  Division 280 

3.  Fractional  Powers  and  Roots  of  Quantities  with  Frac- 
tional Exponents,    a.  Remark 281 

288.  Exercise 281 


TAHLK   OK   CONTENTS.  10 

AKTULK.  CIIAPTEK    XX. —  Of  Kaou  ALS.  ia<;k. 

281).  Definition.     <i.  Katioiuil  Quantities l'S4 

21K),    Radicals  wlien  ]{oots  cannot  be  tak»>n  =  Irrationals  (►r  Sunis  2S4 

((.   Distinction   between  Onlinary  Decimal   Fra<'tions  and 

Irrational  Decimal  Fractions l'S4 

2i)l.    Treatment   of    Radicals,  1.    Reductions.     2.    Fundamental 

Rules.     :].     E<iuations.     a.  Note 2S4,  28.'> 

Skction  I.  —  Heduction  of  liadinils. 
2'.)_*.   Kinds  of  Keduction    . 285 

I.  — SIMI'LIFICA  rioNS. 

293.   Reduction  of   Radicals  to  Efiuivalcnt  ones  liavin*,'  a  Lower 

Index 28."> 

2{)4-21)o.  Rule.     Exercise 285 

29t>-298.  Simi)lification  by  Removing:  a  Factor  from  tbe  Radical. 

Rule  1-2.     Exercise 280,  287 

299-301.  Simplification  of  Fractions  under  tlic  Si<;n.     a.  Reason. 

Rule  l-:i.     Exercise 287,  288 

II.  —  KKDl'CTION    TO    STHD    F()«M.       CONVKIJSK    OI'KIJATIOX. 

302-304.  Reduction  of  EIntire  Quantities  to  the  Form  of  Surds. 

Rule.     Exercise 289 

305-307.    Reduction  of    Ra<licals  to  the  Fonn  of  Surds  having 

Unity  for  Coerticients.    Rule  1,2.     Exercise 290 

III.  —  KKDICTIOX    OF   RADICALS   TO   TIIK    SAMK    INDKX. 

308-310.  Reduction  of  Radicals  to  a  Common  Index.     Rule  1, 

3.     Exercise 291,  292 

Section  II.  —  Fundamental  Opcrathnts  with  Radicals. 

311-Jil3.    Addition   and    Subtraction   of    Radicals,     a.  Answers 

in  Simplest  ?\)rms.    Rule  1,2.    Exercise       ....     292,293 

314-^^10.  Multiplication  and  Division  of  Radicals,  a.  Explan- 
atory.    Rule  1-3.     Exercise 294,  295 

317-^319.    Raticmalization  of  Denominators.      Kule  1-4.     a.  The 

Advantajje.     Exercise 297-299 

320.  Powers  and    Roots  of    Radicals.      Exception    to   (Jeneral 

Rule.     Exercise 300 

321.  Roots  of    Radical    Quantities  not  in  the  Original  Product 

Form.     «.  Explanatory 301 

322.  Theorems  Concerning  Eijuations  Containing  Radicals.     1, 

2,  3 302 

323.  The  Square  Root  of  Binomial  Sunls 30:3 

1.  Demonstration.     2.  Rule.     3.  Examples    ....     :)():{,  304 

324.  Exercise 304 

325-326.     Fundamental     Oi)erations    with    Imaginary    Radical 

Quantities.      Exercise 305-^307 

Section  III. — Equations  Involving  Radicals. 
327-329.    Process  of  Solution.     Rule    ;ui<l    Directions.     1,  2,  3, 

«,  6,  c.     ExeR-ise :',1(M313 


20  TABLE    OF    CO^'TE^'TS. 


FOUKTH   GENERAL   SUBJECT.— QUADKATIC 
EQUATIONS. 

CHAPTER  XXI.  —  Quadratic  Equations  Containing  One 
Unknown  Quantity, 
article.  i'age. 
330.  Definition.    Kinds.   Complete  and  Incomplete,    a.  Explana- 
tory      .  315 

Section  I. — Incomplete  Quadratic  Equations, 

331-332.  Method  of  Solution.     Type   Form.     a.     Two  Hoots  ±. 

Exercise 316-318 

Section  II.  —  Complete  Quadratic  Equations. 

333.  Method  of  Solution 318-320 

334.  The  Coefficient  of  x'^ 320 

335.  Completing  the  Square 320 

336.  Examples 321 

337.  Rules 322 

1.  (l-(3.     2.  (1.    (2.     3.     (1.  (2.     4.  (1.  (2.     Example     .     .  323 

338.  Exercise 324-326 

Section  III. — Problems  of  Quadratic  Equations. 

339.  Problems   of    Complete  and  Incomplete   Quadratic   Equa- 

tions,    a.     Negative  Solutions 326-332 

CHAPTER  XXII.  —  Simultaneous  Quadratics. 

340.  Simultaneous  Quadratic  Equations.     Cases 333 

Section  I. — Simultaneous  Equations,   One  of  which  is  of  the 
Second  Degree,  and  the  Others  Simple  Equations. 

341.  Solution    of    Simultaneous    Equations,   one   being    Quad- 

ratic       333-334 

342-343.  Special  Forms,     a.    Answers  in  Sets.     Exercise  .      334-337 

Section  II.  —  Simultaneous  Equations,  Two  or  more  of  which 
are  of  the  Second  Degree. 

344.  Solution   of   Simultaneous  Equations,  at   least  Two  being 

Quadratic 337-340 

a.    General  Equation  of  the  Second  Degree  in  two  Un- 
knowns        337 

345.  Classes   of   Simultaneous   Equations  having  Two  or  more 

of  the  Second  Degree 340-341 

a.  Methods  not  Mutually  Exclusive 341 

346.  Exercise 341-343 

Section  III.  —  Problems. 

347.  Problems  in  Simultaneous  Quadratic  Equations      ,     .     343-345 


TAliLE   OF   CONTENTS.  21 


(  liArri:i;    win.    -i^iadkatic  K(irATi<)Xs  AM)   Kqiations 

IX    CiKXERAL. 
ARTICLE.  I'AOE. 

348.  Properties  and  Solutions  of  Quadratic  Equations  and  Equa- 
tions in  (ienenil.  Discussion  of  Problems  and  Validity 
of  Methods 346 

Section  I. — Properties. 

MO.  Sum  of  the  Roots  of  a  Quadratic  Etjuation 346 

:i>0.  Pro<luct  of  the  Roots  of  a  Quadratic  Equation      ....  347 

:^">1.  Solving?  a  (Quadratic  Equation  by  Factorin}; 347 

352.  Solvini^E(piationsof  any  Degree  by  Factoring    ......  348 

Ii53-;io4.  Rule  for  Solving  Equations  of  any  Degree  by  Factor- 
ing.    1,  2,  3.     a,  b,  c.     Exercise 349,  350 

355.  Conversely,    to  construct   an   Equation,   having  given   its 

l{(K)ts.     a.     Iniaginaries 350,  351 

Section  II.  —  Discussion  of  Problems.  —  Failure  of  Processes 
of  Solution. 

'M>().  Discussion  of  the  Equation  of  the  Second  Degree.  Great- 
est and  Least  Values  of  the  Coefficients.  Two  Classes. 
1.  Real  Roots,      2.  Imaginary  Roots :^'>2,  353 

357.  Maxima  and  Minima  Values :i53-357 

a.. P^xplicit  and  Implicit  Functions 353 

358.  Discussion  of   the  Equation  x=^  for   limitiii<;  Values  of 

b 
a  and  6.     Infinites  and  Infinitesimals.     1-5     .     .     .     357-359 

.359.  Problem  of  the   Lights        .359-362 

3»K).   Validity  of  Processes  of  Solution 362-1^66 

1.  In  the  Extraction  of  Roots     (1.  (2.  (3 362 

2.  An  Equation  obtained  by  Squaring  or  Cubing,  etc.    (1. 

(2 3(52-.3(J4 

3.  An  Equation  may  not  be  multij)lied  or  divided  through  by 
a  Function  of  the  I'^nknown  which  becomes  either  Zero  or 
Infinity.     (l.-(5 364-366 

Section  III.  — Equations  of  Higher  Degrees  solved  like  Quadratics. 

361.  Equations  containing  but  one  Unknown  solved  like  Incom- 

plete Quadratics 3(MJ-368 

362.  Equations  containing  but  one  Unknown  solved  like  Com- 

l)lete  Quadratics.    Two  TyjHi  Forms,    (ieneral  Remark,  3(W-373 
3(^3.  Simidtaneous  E<iuations  of  Higher  Degrees  whose  Solutions 

are  Reducil)le  to  those  of  Quadratics 373-376 

a.  Number  of  Sets  of  Answers 373 

b.  Verj'  Varied  in  Character 373 

364.  Problems  dei)ending  on  the  Solution  of  Higher  Equa- 
tions   376,  377 


22  TABLE   OF    CONTENTS. 


FIFTH   GENEKAL    SUBJECT.  —  TOPICS   RELATED 

TO  EQUATIONS.  — INEQUALITY,  PROPOETION, 

EXPONENTIAL  EQUATIONS,  LOGAPvITHMS, 

PEOGBESSIONS,  AND  INTEREST. 

ARTICLE.  CHAPTER     XXIY.  — IXEQT'AT.TTIES.  PAGE. 

305.  Definition  of  an  Inequation 378 

a    Signs  of  Inequality 378 

366.  Properties   of    Inequalities 378 

1.  Transposition.    2.  Multiplied  or  divided  by  Positive  Xum- 
ber.  •3.     Powers     of     Members     of     Inequality.      When 

Valid 378-380 

4.  Roots.     When  Valid 380 

.5.  Addition  of  Inequalities 380 

6.  Subtraction  of  Inequalities 380 

7.  Multiplication  of  members  of  Inequalities        380 

367-368.  Special    Theorems,      l.a '-^  + //^>  2a/>.  2  —  110.      Exer- 
cises and  Problems 381 ,  382 

CHAPTER  XXV. —Ratio  and  Pkoportiox. 

369.  Definition  of  Ratio,     a.  Denoted  by  Colon 383 

h.  Other  Way  of  Writing 383 

370.  Reduction  of  Ratios .  383 

371.  Comparison  of  Ratios 384 

372.  Compounding  Ratios 384 

a.  Duplicate,    Triplicate,  etc 384 

h.  May  be  seen  in  Arithmetic 384 

373.  Definition  of  Proportion,     a.  The  Colon  Form  read,  etc.  .  384 

374.  Method  of  Treatment 384 

375.  Transformations  of  a  Proportion 385 

1.  Theorem.     Test.     2.  Converse.     Different  Forms    .     .     .  385 
3.  General  Theorem 385-386 

a.  Inversion 386 

h.  Alternation 386 

Queries  (l.)-(5.)        386 

376.  Xew  Proportions  from  a  Given  one 386 

1.  The  Composition  Theorem,     a.  Direct  Derivation  .     38(),  387 

2.  The  Division  Theorem 387 

3.  The  Composition  and  Division  Theorem 388 

4.  The  Like  Powers  Theorem 388 

5.  The  Like  Roots  Theorem 388 

377.  Combinations  of  Two  or  more  Proportions 388 

1.  In  a.  Continued  Proportion,  etc ,388 

a.  Shorter  Notation 389 

h.   Application  in  Geometry 389 

2.  Multiplication  and  Division  of  the  Corresponding  Terms 

of  Two  or  more  Proportions  ,.,,,. 389 


TAIiLK    <>l     CONTENTS.  -j;} 

:i7S.  Si»ecial  F'onns :;;>0 

1.  A  Moan  l*roiK)rtioual ;J5)0 

<i.   Not  tho  same  as  an  Arithint'tical  >!<•  in ;>;>0 

2.  Two  Mean  Proportionals :^90 

;170.  Exercise  in  Hatio  and  Proportion      .......     :V.)0,  301 

■.\<(y    Variation.     1.  Direct.     2.  Inverse.     ;;.   otiirr.     .     .     :',1)2.  :}i>:5 

(lIAPTKn    XXVI.  —  Kxi'ONKXTiAi.    Im/i  ath>\s    a\i> 

LiXiAltlTIIMS, 

".-^i.    Detinition  of  Exponential  E<|iiations 394 

a.   Failnre  except  in  Ca.se  of  Exact  Powers 3JM 

:)S2.   Lojiaritluns.      a.    Three  Nnnibers  considi'n-d HU4 

h.  Logarithms  approxituate :5!)4 

'.\K\.  Systems  of  Logarithms.     Tables :Ji)5 

a.  Logarithm  of  1  =  0 ;Jl)5 

\.  2  as  the  Base  of  a  System :}S)5 

b.  Logaritlims  are  Fractional  Ex])onents :>{).■> 

2.  Unity  cannot  he  the  Base  of  a  System.     Other  iJascs      .  :Ji)r> 

c.  Base  taken  Positive  ami  Greater  than  1 :W($ 

384.  Tenns  and  Notation 35)6 

:i85.  The  Briggsian   or   ( 'ommon  System 3!M$ 

\.  Logarithms   of  Fractions.     2.    Of  Numbers   between  1 
and  10. 

3.  Of  Numbers  Greater  than  10 :5!><'),  3!>7 

3M»5.   Briggsian  Mantissas 31>7 

(I.  Mantissas  always  taken  Positive 400 

Tables  of  Logarithms :50{),  399 

387.  Explanation  of  accompanying  4-Place  Table, 400 

;i88.  Rules  to  tind  the  Logarithm  of  any  Number 4(X) 

1.  Characteristics 400 

2.  Manti.s.sas 402 

(1.  Mantissas  of  Numbers  of  Three  Figures 402 

(2.  Mantissas  of  Numbers  of  more  than'I'hree  Figures     .  402 

(3.  Exercise 402 

389.  Conversely.      To  tind  a  Number  from  its  Logaritlnns.     To 

find  the  Significant  Figures  from  the  Mantissa     ....  404 

1 .  When  Mantissa  is  same  as  in  the  Table 404 

2.  When  Mantissa  is  not  the  same  as  one  in  Table      .     .     .  404 
3iK).  Exercise 40."> 

391.  Uses  of  Systems  of  Logarithms 405 

392.  To  Multiply  by  Logarithms.     Examj)!*'.  Kule,  Exercise  .     .  40<5 

393.  To  Divide  by  Logarithms.  1.  Example.  2.  Rule.  3.  Exercise. 

4.  Evaluation  of  ( 'omi)ound  Exi)ressions.    5.  ?^\ercise.  407,  408 

394.  To  Hai.se  to  I'owers.     1,  2,  3 408,  401> 

39.5.  To  Extract  Hoots.     1,  2,  3 40i»,  410 

3(m.  Accuracy  of  Results 410,411 

397.  Logarithms  to  Other  Bases  can  be  derived  from  the  Briggs- 

ian Logarithms 411,  412 

398.  Solution  of  Ex]K)nential  Equations  by  Logarithms.    Exam- 

ples.    Exercise 412 


24  TABLE   OF   CONTENTS. 

ARTICLE.       CHAPTER   XXVII. —The  Progresstoxs.            page. 
:]99.  Arithmetical  Progression 414 

400.  Fonimla  for  finding  the  ?i"'  Term  in  a.  j). 414 

401.  Formula  for  finding  the  Sum  of  n  Terms  in  a.  p.        ...  415 

402.  Other  Problems  in  a.  p 416 

408.  Exercise 417-419 

404.  Definition  of  Geometrical  Progression 420 

405.  Formula  for  finding  the  n'"  Term  in  *y.  j> 420 

406.  Formula  for  finding  the  Sum  of  n  Terms  in  g.  p 420 

407.  Other  Problems  in  g.  p. 

1.  Solved  without  Logarithms 421 

2.  Solved  with  Logarithms 422 

3.  Problems  which  lead   to  the  Solution  of  Equations  of 

Higher  Degrees  than  Simple  or  Quadratic 423 

408.  Summation  of  a  Decreasing  g.  p.  which  extends  to  Infinity,  423 

409.  Repeating  Decimals  as  Examples  of  f/.  p.'s 423 

410.  Exercise 424,  425 

CHAPTER  XXVIIL  — Interest,  Annuities,  and  Bonds. 

411.  Interest  and   Annuities  as  Furnishing  Exercise   in  Loga- 

rithms and  the  Progressions 426 

Section  I.  —  Interest. 

412.  Definition  of  Interest.    Kinds 426 

413.  Simple  Interest.     Problems 426 

414.  Annual  Interest 427 

415.  Compound  Interest,  Definition  and  Problems 427 

1.  To  find  the  Amount  when  the  Principal,  Rate,  and  Time 

are  given 428 

2.  To  find  the  Present  Worth  of  a  Sum  Payable  at  a  Criven 

Time  at  a  Given  Rate  of  Discount 428 

3.  To  find  the  Time  when  the  Amount,  Principal,  and  Rate 

are  given 429 

4.  To  find  the  Rate  when  the  Amount,  Principal,  and  Time 

are  given 429 

Section  II. — Annuities. 

416.  Annuities  Certain,  Definitions,  and  Problems 429 

1 .  To  find  the  Amoimt  of  an  Unpaid  Annuity 430 

(1.  Overdue 430 

(2.  Sinking  Fund 431 

2.  To  find  the  Present  Cost  of  an  Annuity 431 

(1.  Annuity  Perpetual 432 

(2.  Deferred  Annuity 432 

(3.  Repaying  a  Loan  in  Annual  Instalments  ....  432 

Section  III.  —  Bonds. 

417.  Definition  and  Problems  in  Bonds 433 

1.  To  calculate  Price  at  Time  of  Issue  to  realize  given  %    .  433 

2.  When  Interest  Payments  are  made  q  Times  a  Year    .     .  434 

4.  Value  when  Payments  have  been  made.  Also  at  Intervals.  436 

5.  To  calculate  Rate  of  Interest  for  Given  Cost      ....  436 

418.  Exercise  in  Interest,  Annuities,  and  Bonds 437 


TEXT-BOOK   OF   ALGEBRA, 


INTRODUCTION. 


CHAPTER    I. 

AI/JKBRA    AS    A     i:i:AN("H    (»F    M  ATHKM  ATICS. 


J>Y  way  ot  preparation  lor  the  study  of  alj^ehra,  it  will 
be  helpful  to  show  its  nature  as  one  of  the  mathematical 
sciences,  and  to  point  out  its  relation  to  two  other  hraiuhes 
of  mathematies,  arithmetic  and  geometry. 

1.  Mathematics  is  the  science  of  the  ex;i«f  relations  of 
quantity  as  to  magnitude  and  form. 

a.  The  word  "  quantity  "  comes  from  a  Latin  adjective  which  means 
"how  much,"  or  "how  many."  Anything  that  lias  size  or  can  he 
measured  is  a  quantity.  Any  area,  as  100  acres,  any  content,  as  25 
bushels,  any  length  of  time,  as  10  liours,  any  number,  as  1.'),  is  a 
quantity. 

Quantity  appears  under  one  or  other  of  two  fonns,  luunber  or 
extent.  Thus  we  may  say  that  a  l)asket  containing;  apples  has  iV2 
in  it;  or  we  may  si^ak  of  the  number  of  inhabitants  in  a  t6wn  as, 
e.g.,  2000.  Or,  on  the  other  hand,  we  may  s]M>ak  of  a  square  mile, 
or  a  cord  of  wood,  and  in  this  way  denote  the  size  of  the  obj<'ct 
named.  Now,  many  quantities  have  not  only  size  but  also  shajKN  e.g., 
a  house  or  a  field;  and  so  mathematics  treats  both  of  tlie  size  of 
objects  and  of  their  8hai>es. 

*  This  chapter  may  be  entirely  omitted  at  tlie  discretion  of  tlie  teacher.  It  is 
too  difficult  for  younp  pupils. 

25 


•20  TEXT-BOOK   OF   ALGEBRA. 

The  following  are  intended  as  suggestive  rather  than 
exhaustive  definitions  of  arithmetic  and  geometry. 

2.  Geometry  treats  of  quantities  in  respect  to  their  posi- 
tion, size,  and  shape. 

3.  Arithmetic  treats  of  numbers  with  reference  to  the 
art  of  computation. 

a.  Arithmetic  shows  how  to  write  numbers  in  the  shortest  and 
most  convenient  way;  how  to  multiply  and  divide  them;  how  to 
extract  roots,  and  the  like.  Algebra,  on  the  other  hand,  treats  of 
numbers  in  the  way  of  finding  general  truths  in  regard  to  them. 

4.  Algebra  is  that  branch  of  mathematics  which  employs 
general  characters  as  well  as  figures  in  the  study  of  numbers, 
marking  its  numbers  off  into  two  opposite  kinds. 

5.  The  General  Characters  used.  —  In  arithmetic  we  study 
numbers  by  using  the  arable  system  of  writing  them. 
In  algebra  not  only  figures  stand  for  numbers,  but  letters 
regarded  as  general  characters  are  used  for  the  same  pur- 
pose, each  letter  standing  for  some  number.  However, 
when  a  letter  is  used  to  stand  for  a  number,  it  is  not  like  a 
figure,  as  5  (which  stands  for  5  only,  and  can  not  mean  any- 
thing else),  but  may  stand  for  any  number.  This  peculiarity 
the  student  will  find  one  of  the  principal  advantages  algebra 
has  over  arithmetic. 

Arithmetic  teaches  how  to  add  and  subtract,  multiply  and 
divide,  extract  roots,  and  the  like,  when  figures  stand  for 
the  numbers;  algebra  teaches  how  to  perform  the  same 
operations  when  letters  stand  for  the  numbers. 

6.  The  Two  Kinds  of  Numbers.  —  The  definition  gives  an- 
other difference  between  arithmetic  and  algebra.  Arith- 
metic employs  only  one  kind  of  numbers,  using  that  kind  to 
stand  in  one  place  for  a  debt,  in  another  for  a  credit ;  some- 
times for  a  gain,  sometimes  for  a  loss,  and  so  on.  Now, 
when  both  gain  and  loss,  for  example,  appear  in  the  same 


INTKODlCTloN.  27 

problem,  contradictions  may  arise  in  the  arithmetical  lan- 
guage. Thus,  if  a  man  in  Imsiness  gain  $i)()0  in  the  first 
part  of  a  year,  and  lose  ^2(K)  in  the  latter  part,  the  amount 
of  his  year's  gain  is  found  by  subtracthiff  $200  from  $500. 
Couti-ariwise,  if  two  men  start  in  business  with  the  same 
sum,  and  one  loses  $20(X),  while  the  other  gains  $3000,  the 
difference  in  their  fortunes  will  be  obtained  by  adiluKj  $2000 
to  $,'5000.  The  same  peculiarity  presents  itself  in  examples 
in  longitude  and  time.  We  ask  what  is  the  difference  in 
longitude  between  Xew  York  and  Berlin,  and  expect  the 
student  to  add  to  get  the  result.  Algebra  makes  such  (pies- 
tions  clear  by  pointing  out  the  opposite  nature  of  the  num- 
bers and  marking  them  in  such  a  way  as  to  show  tliis.  It 
also  enables  one  to  solve  more  difficult  ])roblenis  containing 
such  numbers. 

7.  Opposite  numbers  will  be  investigated  in  the  next 
chapter,  before  taking  up  the  literal  notation.  All  the  laws 
governing  the  simultaneous  use  of  such  lunubers  will  be 
develoi)ed  while  still  using  the  familiar  arabic  system. 


28  TEXT-BOOK  OF  ALGEBRA. 


FIRST    GENERAL    SUBJECT.— THE    ALGE- 
BRAIC  NOTATION. 


CHAPTER    II. 

OPPOSITE    NUMBERS. 

8.  Ill  Algebra  numbers  are  separated  into  two  opposite 
kinds. 

We  will  study  the  nature  of  opposite  numbers  first,  and 
afterwards  consider  the  changes  necessary  in  the  funda- 
mental operations  of  addition,  subtraction,  multiplication, 
and  division. 

SECTION   I. 

Nature  of  Opposite  Numbers. 

9.  Arithmetical  and  Opposite   Numbers.  — The    science   of 

arithmetic  recognizes  only  real  objects  and  will  admit  no 
element  of  unreality.  It  would  say,  for  example,  that  a 
man  can  not  lose  more  than  he  already  has ;  that  there  is 
no  such  thing  as  a  number  less  than  zero.  This  may  be 
described  as  a  failure  in  arithmetical  analysis.  For  it  fre- 
quently happens  that  a  man's  debts  exceed  his  credits,  and 
he  is  worse  off  than  if  he  had  nothing  at  all.  So  stocks 
vary  from  premiums  to  discounts,  and  latitudes  from  north 
of  the  equator  to  south  of  tlie  equator,  and  so  on.  As  was 
pointed  out  and  illustrated  in  Art.  6,  the  operations  of 
addition  and  subtraction  often  become  confused  in  the 
solution  of  ])roblems,  because  no  distinctions  were  made  at 
the  outset  between  the  two  kinds  of  numbers. 


AL(J  K131 : AlC    N OTATIUN . 


29 


10.  The  Arithmetical  Series.  —  Arithmetic  uses 
only  one  set  of  numbers,  commencing  with  1  and 
increasing  without  limit.  We  may  write  down 
an  arithmetical  series  as  follows: 

1  2  S  4  o  6  7  8  9  10  11  12  IS  14  .  ,  .  ^ . 

(See  Art.  66  for  dots  of  continuation,  and  62,  b 
for  the  meaning  of  oo ,  which  is  used  to  denote 
an  indefinitely  large  number.) 

Fractional  and  irrational  numbers  are  included 
between  the  whole  numbers. 

11.  An  Opposite  Series.  —  Let  us  now  commence 
at  zero  and  write  a  similar  series  extending  in  the 
opposite  direction,  understanding  that  every  num- 
ber on  the  left  has  a  signification  opposite  to  that 
of  the  same  number  on  the  right.  If  numbers  to 
the  right  hand  mean  credits,  then  numbers  to  the 
left  hand  mean  debts;  if  numbers  to  the  right 
mean  north,  then  numbers  to  the  left  mean  south ; 
and  so  on.     Combining  the  two,  we  have 


oD  ....  7,  6,  5,  4,  3, 


,  0,  1,  2,  3,  4,  5,  6,  7  ....  00 


12.  The  Series  to  be  used.  —  For  convenience 
merely  we  will  imagine  our  double  series  of  num- 
bers to  extend  vertically.  In  order  to  distinguish 
between  "above"  and  "below"  numbers  when  re- 
moved from  their  places,  some  marks  will  be 
necessary. 

13.  Marking  the  Two  Kinds  of  Numbers.' —  The 
letter  a  (for  above)  might  be  written  over  all  the 
upper  numbers,  and  tlie  letter  b  (for  below)  over  all 
the  lower  ones.     However,  the  notation  adopted 

I  The  student  i.<;  asked  to  prepare  such  a  scale  as  is  found  in  the 
mar/?ln  on  stiff  pasteboard,  and  to  make  const.int  reference  to  it, 
v«rifyin>f  hv  the  scale  all  the  addMiOD?  ftnd  subtraction  given,  until 
quite  Iftuiiliar  w)th  it, 


-.   +  00 


13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 


^0  TEXT-BOOK    OF   ALGEBKA. 

is  immaterial  so  long  as  two  convenient  marks  are  chosen 
and  used  consistently.  We  shall  emjjloy  small  j^Zv^*-  and 
minus  signs,  not  to  denote  addition  and  subtraction,  but 
simply  as  marks  for  distinguishing  the  two  kinds  of  num- 
bers.    A  dagger  and  a  dash  might  just  as  well  be  chosen. 

14.  Instances  of  Opposite  Numbers.  —  This  double  series 
of  numbers  finds  an  application  to  problems  involving 
(1)  A  merchant's  gains  or  losses.  (2)  Income  and  outlay. 
(3)  Latitude  and  longitude.  (4)  Scales,  as  the  thermometer, 
etc.  (5)  Time,  a.d.  and  n.c.  (6)  Attractive  and  repulsive 
forces,  as  positive  and  negative  electricity,  etc. 

15.  Definitions  of  Addition  and  Subtraction  in  Arithmetic.  — 

Let  us  now  examine  addition  and  subtraction  for  a 
single  series  of  numbers.  Addition  means  literally  ^i^^^m^ 
together,  and  is  performed  by  counting  the  total  number  of 
units  in  the  different  numbers  added ;  and  subtraction  means 
finding  a  third  number  which  added  to  the  smaller  of  two 
numbers,  will  give  the  larger.  These  definitions  are  sub- 
stantially the  same  as  those  given  in  the  arithmetic.  Now, 
how  shall  they  be  changed  when  both  series  are  in  use  ? 

16.  Algebraic  Addition.  —  Kemembering  the  opposite  na- 
ture of  the  two  kinds  of  numbers  (11),  let  us  understand 
that  adding  an  "  above  "  number  to  another  means  counting 
upward  that  many  units,  while  adding  a  "  below  "  number 
means  the  opposite,  counting  doivmvard  that  many  units. 

To  illustrate,  let  us  take  four  simple  examples : 
+Q  -{.-^^  =  +11         Explanation.  —When  +5  is  added  to  +6, 
we  count  5  units  up  from  +6  and  have  +11. 

-7  -f  -9  =  -16         Explanation.  —  When  "9  is  added  to  "7, 
we  count  9  units  down  from  "7  and  have  "16. 

Both  these  cases  are  just  as  in  arithmetic.  But,  now,  if 
adding  +5  carries  us  5  units  upward  in  the  series,  there  is 


al(;i:bkaic  notation.  -M 

no  ivasoii  why  the  starting-}K)int  (i.e.,  the  minil>er  to  wliich 
the  other  is  julded)  shoiihl  be  one  i)oiiit  in  the  double  series 
rather  than  another.  Let  us  suppose  the  st:iitini,'-p()int  is 
"3,  to  which  we  are  to  add  "'"5. 

-3 -f- +5  =  -.  Exi'LAXATioN.  —  Starting  at  ~3  and  count- 
ing 5  units  ujjwardy  we  have  "'^2. 

Tiikewise,  adding  a  ''  below  "  number  to  an  ••  above,"  we 
write, 

+6  4-  ~9  =  ~3.  Explanation.  —  Startingat  "♦'6  and  count- 
ing 9  units  downward,  we  come  to  ~3. 

17.   Algebraic  Subtraction  —  Hy    the    definition,  Art.    15, 
the  subtrahend  and  remainder  added  must  etpial  the  minuend. 
We  shall  extend  the  application  of  this  principle  to  algebraic 
numbei-s. 
+7  -^  +8  =  ^IT).     \\\  the  rule  for  addition. 

^IT)  —  +8  ="^7.  C'onse(iuently,  when  '•"8  is  subtracted  from 
+15,  in  order  to  get  "•'7,  one  must  count 
downward  8  units. 

+13  -f  -11  =  +2.     Hy  the  rule  for  atldition. 

'*J  —  "11  =  +13.         Hence  to  .subtract  "11  from  +2  we  must 
count  11  units  upward  to  get  +13. 

Therefore  to  subtract  an  ^^aboce^^  number  count  from  the 
minuend  downward  as  mani/  units  as  there  are  in  the  sub- 
trahend, white  to  subtract  a  " belour^  number  count  from  the 
)uinuend  upward  as  many  uuits  as  there  are  in  the  subt ra- 
it end.  Evidently  this  is  Just  the  opposite  of  the  rule  for 
addition. 

a.  To  subtract  one  number  from  another,  or  to  find  their  difference, 
is  the  same  as  to  find  the  didance  between  them  on  the  scale.  If  the 
minuend  is  above  the  subtraliend,  tlie  difference  is  marked  "above," 
-f ;  if  it  is  below,  the  difference  is  marked  "  below,"  — . 


32  TEXT-BOOK   OF   ALGEBRA. 

18.  Exercise  in  Addition  and  Subtraction.  —  The  student 
is  expected  to  write  down  the  answers  and  give  the  reasons 
as  in  example  1. 

1.  +s  -\-+S  =  +16.  To  add  +8  to  +8  count  8  units  up 
from  +8. 

2.  -11  -\r  +16  =  ?  7.  +8  -  +4  =  ? 

3.  -9  +  +2  =  ?  8.  +6  -  +9  =  ? 

4.  -19  +  -1  =  ?  9.  -3  -  -5  =  ? 

5.  +32  +  -17  =  ?  10.  -17  -  -12  =  ? 

6.  +42  -{--55  =  ?  11.  +26  -  "15  =  ? 

12.    -17  -  +29  =  ? 

19.  Meaning  attached  to  Zero  (0)  in  Algebra.  —  Zero  has 
a  real  place  in  the  double  series,  (12)  —  as  real,  for  ex- 
ample, as  the  zero  point  in  the  thermometric  or  any  other 
similar  scale.  It  has  a  real  place  in  the  latitude  scale; 
viz.,  the  equator :  and  so  in  other  instances.  Consequently 
it  can  appear  just  like  other  numbers  in  algebra.  Thus 
adding  +5  to  0  means  going  up  from  0  to  +5.  Adding  "13 
to  0  means  passing  down  from  0  to  ~  13.     We  have 

1.  0  -}-  +6  =  +6.  Starting  at  0  and  moving  up  6  units 
gives  +6. 

2.  0  —  +o  =  "5.  Starting  at  0  and  moving  down  5  units 
gives  "5. 

3.  0  -|-  "3  =  "3.  Starting  at  0  and  moving  down  3  units 
gives  "3. 

4.  0  —  "4  =  +4.  Starting  at  0  and  moving  up  4  units 
gives  +4. 

20.  The  Use  of  Other  Signs  to  Mark  the  Series.  —  From 
the  first  and  second  equations  of  the  last  article  we  see  that 
instead  of  the  sign  "  +  ",  "  ^  +  "^ "  ii^^y  be  used ;  and  instead 
of  the  sign  ''  -  ",  "  0  —  +  "  may  be  used. 


ALGEBRAIC   NOTATION.  o3 

IJut  the  latter  notation  lias  "^  before  both  kinds  oi  num- 
bers. Consequently  this  mark  need  not  be  retained.  Hence 
instead  of  the  sign  "  "•"  ",  "  0  +  "  may  be  used,  and  instead 
of  the  sign  "  ~ '',  "  0  —  "  may  be  used. 

Note. — llatl  the  thirtl  and  fourth  equations  Ixmmi  selected,  the 
results  would  simply  duplicate  those  found,  the  +  and  -  merely  being 
interchanged. 

21.  The  0  in  the  Series  Sign  just  obtained  is  dropped.  — 
Since  0  adds  no  value  and  can  be  readily  supplied,  it  is  plain 
that  it  need  not  be  written,  and  that  its  absence  will  make 
no  difference  in  either  our  operations  or  results.  As  in  the 
first  e(puition  of  Art.  19,  no  sign,  before  a  number  and  a 
-|-  sign  signify  tlie  same  thing. 

To  illustrate, 

0  -h  21  =  +  lil  =  21  ;  0  -  21  =  _  21. 

22.  Conclusion.  —  15y  the  last  two  articles  it  appears  that 
we  nuiy  distinguish  between  the  two  series  by  +  and  — 
signs.  A  -|-  sign  prefixed  to  a  number  shows  that  it  belongs 
to  one,  and  a  —  sign  that  it  belongs  to  the  other  series. 

23.  From  what  we  know  of  the  nature  of  addition  and 
subtniction,  the  sign  of  addition  (-[-)  refers  to  what  may  be 
called  the  direct  series,  and  the  sign  of  subtraction  (  — ),  to 
the  other,  which  may  be  called  the  reverse  series.  Naturally 
no  sign  at  all  is  direct,  so  that  both  +  and  no  sign  refer  to 
this  series. 

24.  Two  Views  of  the  Signs.  —  The  two  signs  -{-  and  — 
seem  to  have  a  double  meaning :  +  to  denote  the  operation 
of  addition,  and  to  mark  direcrt  nund)ers ;  and  —  to  denote 
subtraction,  and  to  mark  reverse  numbers. 

In  the  thermometer  scale  -|-  means  above  zero.  In  ac- 
counts -f  would  ap[)ropriately  refer  to  income,  —  to  outlay. 
But  it  woidd  be  better  to  regard  them  as  always  indi- 
cating the  o})erations  of  addition  and  subtraction,  0  being 
understood  when  no  other  quantity  precedes  either  of  them. 


34  TEXT-BOOK    OF    ALGEBRA. 

25.  Positive  and  Negative  Numbers.  —  lumbers  from  tne 
direct  series  are  called  positive  or  plus  iiuinbers,  while  those 
from  the  reverse  are  called  negative  or  minus  numbers. 

26.  Like  and  Unlike  Signs.  —  When  numbers  or  terms 
have  the  same  sign  preceding  them  they  are  said  to  have 
like  signs ;  otherwise  they  are  said  to  have  unlike  signs. 

Thus  5,  +7,  +13  have  like  signs.  Also  —4,  —3,  —41, 
—50  have  like  signs.  Again  —  28  and  +  4  have  unlike 
signs;  and  3,  —6,  +11,  —23,  —24  have  unlike  signs. 


SECTION    11. 
The  FuNnAMEXTAL  Rules  fou  Opposite  Nimhehs. 

27.  Rules  for  all  of  the  fundamental  operations  will  now 
have  to  be  investigated. 

I. -ADDITION. 

28.  Addition  of  Positive  and  Negative  Numbers.  —  There 
are  two  cases. 

1.  When  the  numbers  added  have  like  signs.  The  oper- 
ations now  to  be  considered  are  the  same  as  those  in  18, 
but  large  +  and  —  signs  take  the  place  of  those  there  used 
to  mark  the  series. 

Explanation   of    (2.  —  By 
16  when  —  7  is  added  to  —  3, 
we  count  7  units  down  to  —  10  ; 
(-'■•        ~\'  ^  (2-     —    2     —  5  added  to  this  result  gives 

—  15;    and    —  12    added    to 

—  15  gives  the  answer,  —  27. 
Or,  more  simply,  the  numbers 
are  added  as  in  Example  (1, 
and  the  —  sign  prefixed  to  the 
result. 


+  4 

(2. 

-    3 

+  6 

—    7 

+  7 

—    5 

+  ^) 

-12 

+  26 

-  27 

al<.i:i;i;aic  NoiwrioN.  35 

(.3.    Set  down  as  tlie  above  iind  add  -|-".),  -j-7,  -}-<),  -\-\i>, 
and  +81. 

(4.    A«ld  -(•)!,  -L':;.  -1>7.  -.").  -1.  and  -IT)!. 

(.\  Add  -s;;,  -8.S,  -s;;,  -s:{,  -8:i,  -8.s,  -8;5,  -S'A. 

2.    W'lien  the  numbers  added  have  unlike  signs. 
(1.    Add  5,  -  4,  +  IT).  -  I),  and  +  1. 

;-,         KxpLAXATiox.  —  Following  the  method  given 

—  4     in  1(),  we  have, 

-|-  15  —    4  added  to  +    5  gives  4-    1? 

—  9  +15  atlded  to  +    1  gives  +  16, 
1  —    0  added  to  +  1(5  gives  +    7. 

8  +    1  added  to  +    7  gives  +    8.    Ans. 

(2.   Add  -  11.  -  :i.  4-  ir>.  -  (;.  and  -  14. 

—  11  Exri-AN  ATIO.N. 

—  o  —    .S  added  to  —  11  gives  —  14. 
+  IT)  +  15  added  to  —  14  gives  +    1. 

—  6  —    0  added  to  +    1  gives  —    5. 

—  14  —  14  added  to  —    5  gives  —  19.   Ans. 
-^19 

(;i    Add  in,  -  0,  ;^,  -  49,  -  LV),  14,  and  2. 

a     Add  ()02.  27.  -  1903.  1292,  -  3r)9,  -  1. 

ll  i>  a  well-known  })rinei|)le  in  aiitlmietie  that  the  sum  is 
the  same  in  whatever  order  the  numbers  are  added.  If  this 
be  true  in  algebra  there  will  be  an  obvious  advantage  in 
changing  the  order  of  the  numbers  added,  so  that  all  the 
l)ositive  numl^ers  come  first,  followed  by  all  the  negative 
numbers.  In  the  examjdes  just  given  it  is  j)lain  that  start- 
ing with  zero  we  count  in  the  i)ositive  direction  in  all  a 
certain  distance,  denoted  by  the  sum  of  the  positive  numbei-s, 
and  in  the  negative  direction  in  all  a  certain  distance  denoted 


S6  TEXT-BOOK   OF   ALGEBRA. 

by  the  sum  of  the  negative  numbers :  so  that  the  sum  ob- 
tained by  adding  the  numbers  seriatim,  or  one  after  another, 
is  the  same  as  that  obtained  by  adding  the  sum  of  the  posi- 
tive numbers,  and  the  sum  of  the  negative.  In  other  words, 
tke  order  in  which  numhej^s  are  added  in  algebra  is  indiffer- 
ent.    This  is  called  the  commutative  law  in  addition. 

Thus,  in  examples  (1  and  (2  above 
(1.    5  +  15  +  1  =  21 ;  -  4  -  9  =  -  13 ;  21  -  13  =  8, 

answer  as  before. 
(2.    15;  -  11  -  3  -  6  -  14  =  -  34;  15  -  34  =  -  19, 

answer  as  before. 

Remark.  —  The  student  has  no  doubt  observed  that  when  two 
opposite  numbers  are  added  the  sum  is  their  numerical  difference.  It 
is  positive  if  the  positive  number  is  the  greater,  and  negative  if  the 
negative  number  is  the  greater. 

The  student  may  verify  the  truth  of  the  commutative  law 
for  exercises  (3  and  (4. 

(5.   Add  -6,-7,  11,  15,  -  3,  -  21,  +  4. 

(6.    Add  19,  -  3,  -  6,  0,  -  4,  -f-  6,  -  31,  and  50. 

29.  Rule  for  Adding  Positive  and  Negative  Numbers. 

1.  If  the  numbers  to  be  added  have  like  signs,  add  them 
as  in  arithmetic,  and  prefix  the  common  sign. 

2.  If  the  numbers  to  be  added  have  unlike  signs,  add  the 
positive  numbers  and  the  negative  numbers  separately,  take 
the  difference,  and  prefix  the  sign  of  the  greater. 

30.  Exercise  in  the  Addition  of  Positive  and  Negative  Num- 
bers.    Problems. 

1.  Add  51,  96,  -  37,  -  72,  101,  -  49,  -f  237. 

2.  Add  -  79,  -  106,  -  304,  +  40,  -  9,  +  1,  and  362. 

3.  142  _  16  -  50  -  31  -  199  +  777  =  ? 


AL(;KBUAir    NOTA'IMOX.  37 

4.  18  X  47  -  39  X  47  -  21  X  47  +  47  =  ? 

5.  Addoi,  -3H,8i,  -193,  _2Ql. 

6.  29.4  -  3.3  +  .079  -  1.001  -  20  +  43.05  +  11  =  ? 

7.  A  boy  has  15  cts.,  his  father  owes  him  12  cts.,  and  one 
of  his  playmates  7  cts.  If  credits  be  marked  +,  how  much 
has  the  boy,  and  how  shall  the  amount  be  marked  ? 

8.  William  has  no  money,  and  he  owes  a  grocer  8  cts.,  a 
l)ieman  4  cts.,  his  sister  3  cts.,  and  a  playmate  10  cts.  What 
is  he  worth,  and  how  shall  we  mark  the  amount? 

9.  A  ship  starting  from  17®  north  latitude  goes  one  day 
2°  north,  the  next  3°  north,  and  a  third  4°  south ;  what  lati- 
tude is  she  now  in,  north  latitudes  being  marked  -j-  and 
south  latitudes  —  ? 

10.  A  vine  that  was  30  inches  long  grew  in  one  month  9 
in.,  in  the  next  12  in.,  and  in  the  next  15  in.  It  was  pruned 
back  at  one  time  11  in.,  and  at  another  time  5  in.  What 
is  its  length  ? 

11.  A  boy  has  25  cts.  and  owes  10;  what  is  he  worth? 
How  shall  25  be  marked,  how  10,  and  how  the  result  ? 

12.  George  has  16  cts.  in  his  money  box,  10  in  his  pocket, 
and  William  owes  him  10  cts.  He  owes  his  brother  John 
7  cts.,  and  a  confectioner  14.     What  is  he  worth  ? 

13.  A  thermometer  which  stood  at  4  p.m.  at  70°,  and 
which  by  8  o'clock  had  fallen  14°,  and  by  midnight  8°  more, 
and  by  4  a.m.  3°  more,  rose  from  4  o'clock  to  8  o'clock  12°, 
and  from  8  o'clock  to  12  o'clock  noon  25°.  Where  did  the 
mercury  stand  at  the  last-named  hour  ? 

14.  A  man  who  has  $300  in  the  bank  and  $125  in  his 
lK)cket  owes  A  $550  and  B  $1000.  He  has  due  him  from 
one  party  $400,  and  from  another  $640.  But  he  owes  as 
surety  on  a  note  $190.     What  is  he  worth  ? 


38  TEXT-BOOK   OF  ALGEBRA. 

15.  A  rope  which  was  40  ft.  long  had  at  one  time  8  ft.  cut 
off,  at  another  6  ft. ;  it  was  spliced  with  a  piece  15  ft.  long, 
after  which  9  ft.  was  first  broken  off,  then  11  ft.,  then  3  ft., 
when  it  was  again  spliced  with  a  20  ft.  piece.  How  long 
was  the  rope  then  ? 

Query.  —  If  a  question  like  this  were  propounded,  in  which  the 
parts  broken  off  taken  together  exceeded  the  original  length  of  the 
rope  added  to  the  sum  of  the  splices,  what  would  the  student  judge 
concerning  the  nature  of  the  problem  ?  What  then  will  a  negative 
result  sometimes  indicate  ? 

II.  -  SUBTRACTION. 

31.  Subtraction  of  Positive  and  Negative  Numbers.  — 
See  17. 

1.        18        2.         15  ExPLANATiox.  —  To  subtract  —  8, 

7  — 8       since  it  is  a  negative  number,  count 

11  23       from  15,  8  units  up  to  +  23. 

ExPLANATiox.  —  To    Subtract 
-J  ^v  Z  'i  -         +  l^j  count  10  units  down  from  the 

minuend.     To  subtract  —  17,  count 


-24  -  1() 


17  units  up  from  the  minuend. 


5.      T)  6.         61  7.       -  3  8.       -  1 

11  -89  +  7  -  12 

-  5  150  -  10  +11 

By  Arts.  16  and  17  we  perceive  that  subtracting  a  num- 
ber of  one  series  is  always  the  same  as  adding  the  corre- 
sponding number  of  the  other  series.     Thus : 

+18  -  +10  =  +8,     as  also,  +18  +  "10  =  +8 ; 
+21  -  -  12  =  +33,     as  also,  +21  +  +12  =  +33 ; 

and  so  in  every  case. 

Now,  since  this  is  true,  if  we  choose  to  do  so,  we  can  sub- 
tract by  first  changing  the  series  to  which  the  subtrahend 


ATX5EKUAIC   NOTATION.  39 

belongs,  and  then  nddinfj  it  to  the  minuend.  But  to  change 
the  series  of  a  nunibL*r,  we  merely  change  its  sign.  Coiis^*- 
quently  we  may  use  this  rule : 

To  Subtract,  rhanf/e  the  sign  of  the  subtrahend  and  add. 

a.  Instead  of  actually  chan£jln<;  the  sign  of  the  subtrahend,  it  is 
more  convenient  to  imagine  it  to  be  done. 

Thus,  in  Ex.  3,  imagine  the  sign  of  10  to  become  — ;  and  then,  if 
—  10  is  added  to  —  14,  the  sum  is  —  24.  In  Ex.  «,  conceive  —  8i)  to 
become  +  8i),  and  adding  +  89  to  01,  we  have  +  l.'iO ;  and  so  in  the 
other  examples. 

9.  From  IG  take  23.  10.    From  —  15  take  —  38. 

11.  44  —  10()  =  ?  12.    From  -  399  take  183. 

13.  From  —  1000  take  1001. 

14.  From  436  take  :m. 

32.  Rule  for  Subtracting  Positive  and  Negative  Numbers.  — 
Conceive  the  sign  of  tlic  subtrahend  nuniln'i-  to  be  cluinged, 
and  add  it  to  the  minuend. 

33.  Exercise  in  Subtracting  Positive  and  Negative  Num- 
b(r8.  — 

1.    From  1428  take  —  794.       2.    From  -  37  take  —  50. 

3.  From  the  sum  of  16,  32,  -  79,  -  37,  -  109.  2,  and  9, 
take  the  sum  of  -  19,  -  108,  -  42,  83,  2i),  -  (;o.  aii.l  41. 

4.  From  19  X  99  take  73  X  99. 

5.  From  16  -  31  -f-  172  -  200  less  1(;9  -  Ty^^^^  -  263 
take  3<)()  -  243  +  27  less  100  -  6  +  23. 

6.  On  a  certain  day  the  mercury  in  a  thermometer  stood 
at  60°,  and  on  the  next  it  stood  at  77°.  Wliat  was  the  dif- 
ference of  temperature  of  the  two  days  if  -j-  denote  u})ward 
movement  and  —  downward  ? 

7.  A  man  who  has  property  valued  at  $3000  owes,  in 
various  amounts,  to  the  extent  of  S3355.  What  is  he  worth, 
and  how  shall  we  mark  the  number  ? 


40  TEXT-BOOK   OF    ALGEBRA. 

8.  What  is  the  difference  in  longitude  between  New  York 
and  Berlin  if  New  York  is  74°  0'  24"  west  and  Berlin  is 
13°  23'  east?  How  shall  we  mark  the  result  if  east  be  + 
and  west  be  —  ,  the  answer  thus  showing  how  far  and  in 
which  direction  New  York  is  from  Berlin  ? 

9.  If  north  latitude  be  marked  +  ^-nd  south  — ,  what  is 
the  difference  in  latitude  between  two  ships,  one  at  19° 
south  and  the  other  at  61°  north  ? 

10.  What  is  the  difference  between  the  average  tempera- 
tures of  January  and  December  if  that  of  December  be  15° 
above  zero  and  chat  of  January  4°  below  ? 

III.  -MULTIPLICATION. 

34.  Multiplication  in  Algebra  is  the  process  of  taking  one 
number  as  many  times  as  there  are  units  in  another,  giving 
the  product  the  same  sign  as  the  multiplicand's,  or  the 
opposite,  according  as  the  multiplier  is  a  direct  or  reverse 
number. 

To  illustrate  (we  suppose  the  first  factor  to  be  the  multi- 
plier) : 

+  5  X  +  6  =  30 ,  _  3  X  +  T  =  -  21 ; 

+  4  X  -  9  =  -  36 ;        _  2  X  -  8  =  +  16. 

a.  The  definition  of  the  multiplication  of  algebraic  numbers  given 
above  is  based  on  the  nature  of  such  numbers.  One  series  (10) 
attaches  a  direct  meaning  to  all  its  numbers,  and  is  the  first  and  nat- 
ural series.  The  other  is  joined  to  the  first,  and  obtains  its  mean- 
ing from  the  first  by  always  using  its  numbers  in  a  sense  contrary  to 
that  which  is  applied  to  those  of  the  first.  Looking  at  the  two  series 
as  written  in  11,  the  two  parts  are  the  same,  or,  rather,  they  are 
symmetrical,  like  the  two  arms  of  a  balance.  Nevertheless,  the 
meanings  attached  to  them  place  them  on  a  different  footing.  So 
we  find,  while  the  product  of  two  positive  numbers  is  a  positive  num- 
ber, that,  owing  to  the  difference  in  their  significations,  the  product 
of  two  negative  numbers  is  not  a  negative,  but  a  positive  number. 
This  last  is  not  imlike  the  grammatical  rule  which  says  two  nega- 


AI.(;KI!I:.\I("    Nr>TATloV.  41 

lives  make  an  atlirniutive.  Thus,  I  said  "not  unlike,"  when  I 
meant  "  alike,'"  at  U*ast,  in  some  respects.  The  problems  in  41 
illustrate  the  definition. 

35.  Investigation  of  the  Rule  of  Signs  in  Multiplication. — 
It  can  be  showu  Iroiu  the  definitions  of  addition  and  sub- 
traction that  if  a  positive  multiplier  gives  to  the  product 
the  same  sign  as  the  multiplicand's,  a  negative  multiplier 
will  give  opposite  results. 

1.  When  the  sign  of  the  niultiidier  is  -f-  • 
(1.    To  multiply  +  15  by  +  7. 

Once  -{- W  is  -\-lo;  twice  +  !•">  is  +  30;  three 
times  -j-  15  is  +  45,  and  so  on.  7  x  +  15  is  +  105. 
And  so  in  general. 

(2.    To  multiply  —  9  by  +  8. 

Once  —  9  is  —  9 ;  twice  —  9  is  —  18 ;  three  times 
—  9  is  —  27,  and  so  on.  8  X  —  9  is  —  72.  And  so 
in  general. 

2.  When  the  sign  of  the  multiplier  is  —  : 

In  the  expression  6  X  12  —  13  x  12  it  is  evidently  in- 
tended that  13  times  12  sliall  be  subtracted  from  G  times 
12,  and  the  difference  is  therefore  —  7  times  12. 

Now, 

6  X  12  -  13  X  12  =  72  -  156  =  -  84.     (Art.  32.) 
Hence,     -  7  x  12  =  -  84. 

Again,  let  us  evaluate  the  expression  3  x  —  14  —  8  x  —  14, 
which  plainly  means  that  8  times  —  14  is  to  be  taken  from  3 
times  —  14,  and  the  remamder  is,  therefore,  —  5  times  —  14. 

But, 

.S  X  -  14  -  S  X   -  14  =  -  42  minus  —  112  =  +  70. 

(Art.  32.) 
Hence,     —  5  x  —  14  =  -f-  70. 


42  TEXT-BOOK  OF   ALGEBRA. 

This  reasoning  is  plainly  applicable  to  any  other  numbers. 

If,  now,  +  refer  to  the  direct  series  and  —  to  the  re- 
verse, these  four  products  agree  with  the  definition  (34). 
But  the  last  two  have  been  obtained  without  any  further 
reference  to  that  article.  Consequently,  we  are  assured  that 
the  definitions  of  multiplication  and  addition  and  subtrac- 
tion are  consistent. 

36.  Rule  for  Signs  in  Multiplication.  —  Like  signs  give 
+,  and  unlike,  —  .  Or,  more  specifically,  -|-  by  +  and  — 
by  —  give  -1-  ;  while  +  by  —  and  —  by  -|-  give  — . 

37.  Exercise  in  Multiplication  of  Positive  and  Negative 
Numbers. 

1.    8  X  -  2  =  ?  2.    6  X  -\-36  =  ? 

3.    -  19  X  3  =  ?  4.    42  X  -  1  =  ? 

5.  _  72  X  -  4  =  ?  6.  2.V  X  -  :n  =  ? 


7.    14|  X  -^  =  ?  8.    4.6  X  -  .23  =  ? 


38.  The  Commutative  and  Associative  Laws  in  Multiplica- 
tion. —  We  knew  in  Arithmetic  that  it  made  no  difference 
in  what  order  numbers  were  multiplied;  for  example, 
7X9  =  9X7,  and  3x4x5  =  3x5x4  =  4x3x5, 
and  so  on.  We  will  show  that  this  is  true  in  ordinary 
Algebra  as  well. 

1.  The  Commutative  Law.  —  That  the  numerical  value 
of  the  product  of  two  factors  is  the  same,  whichever  be  the 
multiplier,  is  known  from  Arithmetic.  We  are  to  show  that 
interchanging  the  factors  will  not  change  the  sign  of  the 
product. 

(1.  If  the  Factors  are  both  Positive  or  both  Negative. — 
In  either  case  the  product  would  be  +?  both  before  and 
after  changing  the  factors  (36). 


AL(;khi:ai<'  notation.  43 

(2.  If  one  Factor  is  Positive  and  the  other  Negative.  — 
Here  the  pn^duct  would  be  nejjative.  both  before  and  aftei- 
ehaiigiiig  (36). 

Hence,  the  multiplier  and  iiiulti})licand  may  change  places 
in  Algebra,  and  the  product  remains  the  same. 

2.  The  Associative  Law.  —  To  see  whether  the  same  law- 
will  hold  for  three  or  more  factors,  let  us  first  take  three 
factors.  Now,  bv  lUiiting  any  two  of  them  into  one  factor, 
we  have  this  prodiu^t  nudtiplied  by  the  third,  in  which 
multiplication,  as  well  as  in  the  first  partial  one,  the  order 
may  be  disregarded  by  the  commutative  law.  And  so  also 
for  a  larger  number  of  factors. 

Hence,  it  may  be  inferred  that  the  product  of  any  number 
of  factors  is  the  same  in  whatever  order  they  are  associated. 

39.    The  Sign  of  the  Product  for  Three  or  more  Factors. 

The  rule  for  signs,  where  two  fa('tors  are  multiplied,  has 
just  been  given.  The  rule  for  a  greater  number  of  factors 
is  derived  from  that  for  two  in  the  following  manner: 

1.  The  product  of  any  number  of  j>ositive  fju'tors  is 
l)ositive. 

For,  any  two  multiplied  together  give  a  positive  product 
(36),  and  this  product  by  a  third  factor  is  ])ositive;  and  so 
on  for  any  number  of  factors,  -f-  by  -(-  always  giving  a 
lK)sitive  product. 

Thus,     -h  r.  X    -f  '•>  X  4-  -  X  4-  1  =  -f  (I  X  9  X  L'  X  I 

=  -f  U)K 
For,        -|-r»x-|-l)  =  -fr>4;  +r>4  x +2  = +108;  +  lOS  x 

+  1  =  +  KKS. 

2.  The  product  of  any  creji  nund)er  of  negative  factors  is 
always  positive. 

If  there  he  positive  factors,  change  the  order  so  jus  to 
bring  them  all  together  (38).     Then,  by  taking  tlie  negative 


44  TEXT-BOOK   OF   ALGEBRA. 

factors  in  pairs,  since  two  negative  factors  give  a  positive 
product,  these  positive  products  with  the  positive  factors,  if 
any,  give  a  positive  product  by  1  above. 

Thus,     -6x-3x-2x  -5x  +4x  +l=+18x 
-f  10  X  +  4  =  +  720. 

3.  The  product  of  an  odd  number  of  negative  factors  is 
negative. 

For,  if  one  negative  factor  be  withdrawn  (since  one  less 
than  an  odd  number  is  an  even  number),  the  product  then 
obtained  from  the  others  will  be  positive  by  the  preceding 
case.  This  positive  product  multiplied  by  the  negative 
factor  withdrawn  gives  a  negative  result  (36). 

Thus,     +8x  +2x  -7  =  16x  -1  =  -  102. 

Also,  +5x— 3x-9x— 4x  — 2x-4  =  4-5x 
+  27x+8x-4=+  1080  x  -  4  = 
-  4320. 

40.  Rule.  —  If  the  number  of  negative  factors  in  a 
product  be  odd,  the  product  is  negative.  Otherwise,  it  is 
positive. 

41.  Exercise  in  the  Multiplication  of  Opposite  Numbers. 

1.    6  X  11  X  -  25.  2.    19  X  -  1  X  -  1  X  -  2. 

3.  72  X  2  X  -  1  X  -  40. 

4.  11  X  -  4  X  -?  X  --. 

3  3 

5.  _  .6  X  -  .G  X  -  .3  X  -.02. 

2  4  3  0 

6.  -X-^X-'^X-^. 

3  9  8  5 

7.  11  X  -  3  X  -  4  X  -  7  X  -  21  X  2. 

8.  -Ix— Ix— Ix— Ix— 2x2x2x-3x 

-  3  X  3. 

9.  -6=-5x-4x-3x-2x-lxlx2 

X  3. 
10.    6  X  7  X  2  X  3  X  4  X  17  X  2  X  -  1. 


ALGEBRAIC   NOTATION.  45 

11.  A  boy  engaged  in  catching  fish  secures  each  day 
50  tish  for  10  successive  days.  How  many  did  he  catch  in 
all  ? 

12.  A  hunlt'i-  ustHJ  24  charges  of  ammunilion  without 
securing  any  game,  each  charge  costing  him  2^  cents.  If 
waste  be  marked  — ,  how  shall  we  write  the  factors  and 
product  indicating  the  expense  connected  with  his  sport  ? 

13.  A  manufacturer  of  telescopic  object  glasses,  the 
casting  of  which  costs  $150  each,  out  of  a  certain  number 
cast  breaks  11  in  the  grinding.  If  the  number  of  those 
broken  be  marked  — ,  how  shall  we  indicate  his  loss  in  the 
factors  and  product  ? 

14.  *  *  The  ten  hindmost  cars  of  a  train  which  is  going 
directly  across  a  valley  are  still  directed  down  hill,  and 
besides  overcoming  friction  are  exerting  a  forward  tendency 
equal  to  two  horse-power  each.  How^  shall  we  represent 
their  combined  effect  on  the  train  by  using  numbers  from 
the  double  series  ? 

Solution.  —  On  the  level  the  force  exerted  by  the  loco- 
motive in  a  forward  direction  would  ai)propriately  be 
marked  -f ,  while  the  resistance  of  each  car  would  be 
marked—.  On  this  supposition  we  mark  every  car's 
effect  — ,  and  in  the  case  of  the  ten  hindmost  each  —  2. 
But  the  ten  hindmost  cars  act  contrary  to  the  original  sup- 
position.    Therefore  we  write 

—  10  X  —  2  horse-power  =  -|-  20  horse-[K)wer ; 

i.e.,  20  hoi*se-iX)wer  in  the  jiositive  direction. 

15.  ♦  *  A  l><»)k-agent  selling  a  book  at  $(J.50  which  costs 
him  $4  finds  him.self  at  the  end  of  a  certain  year  in  debt  to 
his  company  for  24  copies,  the  pay  for  which  he  never 
expects  to  be  able  to  collect.  The  next  year  he  sells  150 
copies  of  a  book  at  $5  a  co])y,  the  cost  of  which  is  $.*>. 
r>ut  at  the  end  of  this  vear  he   Hnds  he  has  been  able  to 


46  TEXT-BOOK   OF   ALGEBRA. 

collect  for  only  148  copies.  During  the  year,  however, 
9  copies  of  the  first  book  have  been  paid  for  leaving  only 
15  still  unpaid.  It  is  required  to  represent  his  gains  and 
losses  by  factors  and  products  from  the  double  series. 

Solution.  —  In  his  accounts  the  agent  would  naturally 
keep  the  old  and  the  new  items  separate.  The  former  as 
supposed  loss  would  appropriately  be  marked  — ,  and  the 
latter  as  supposed  gain  +.  Again  the  number  of  books 
paid  for  and  the  number  of  books  not  paid  for  i7i  accoi^dance 
with  his  expectations,  would  be  direct  series  numbers,  while 
the  number  of  books  not  paid  for  and  the  number  of  books 
paid  for  contrary  to  his  expectations  would  be  reverse  series 
numbers 

+  15  X  -  $4  =  -  $60 loss;  +  148  X  $2  =  +  $296  gain; 

-  12  X  $3  =  -  $36  loss ;  -  9  X  -  $2.50  =  +  $22.50 
gain. 

Adding,  -  $60  +  $296  -  $36  +  $22.50  =  $222.50  net 
profit. 

42.  Powers  of  Algebraic  Numbers.  —  The  term  power  of 
a  number  is  used  in  the  same  sense  in  algebra  as  in  arith- 
metic. 

Thus,    62  =  6X6  =  36;    (-2)*  =  -2x-2x-2x 

-  2  =  +  16. 

a.  In  algebra  it  is  often  convenient  to  place  the  exponent  (i.e., 
the  small  figure  written  to  the  right  and  above  the  number)  outside 
a  parenthesis,  as  in  the  second  example  just  given.  This  means  that 
the  number  inside  with  its  proper  sign  is  to  be  taken  as  a  factor  as 
many  times  as  the  exponent  has  units. 

1.    The  Signs  of  Powers. 

(1.  If  we  use  a  positive  number  continuously  as  a 
factor,  we  always  get  a  positive  product  (39).  Hence  any 
power  of  a  positive  number  is  also  positive. 

(2.  Even  powers  of  negative  numbers  are  positive 
(39,2). 


AICKfiKAir    NOFATIOX.  47 

(3.    Odd  powuiij  ot    negative    numbers   are    negative 
^39.  S). 

2.    Exercise  in  raising  nunihi'is  to  powers. 

(1.  Square  16. 

(2.  Raise  -|-  4  to  the  fourth  power. 

(3.  Cube  16. 

(4.  Raise  —  3  to  the  fifth  power. 

(5.  Cul)e  —  14.  (6.    Square  —  11.1. 

(7.  (-O)'*^?  (8.    (-2)«. 

(9.  (Shy.  (10.    (-5X  -3)*. 

(11.  (3  X  -  2'^y. 


IV.  -  DIVISION. 

43.  Division  in  Algebra  is  the  process  of  finding  one 
factor  when  the  })roduct  and  tlie  other  factor  are  given. 

The  product  is  the  dividend,  the  given  factor  is  the 
divisor,  and  the  required  factor  is  the  quotient. 

Since  division  is  thus  seen  to  be  the  converse  of  multipli- 
cation, the  law  of  signs  in  division  may  be  inferred  from 
that  in  niuUii>1i<-;ition. 

By  thf  (l»Miiiii»oii  «)f  division,  it 

^-  6  X  4-5=4-  30,  then  4-  30  -^  4-  (>  =  4-  5;   i.e., 

4-  divided  by  4-  gives  4-. 
4-  <)  X   —  5  =  —  30,  then  —  30  -i-  4-  6  =  —  5 ;   i.e., 

—  divided  by  4-  gives  — . 

_  (j  X  4-  5  =  —  30,  then  —  30  ^ 6=4-5;    i.e., 

—  divided  by  —  gives  4-. 

—  Ox  -  5  =  4-  «^,  then  4-  30  ^  -  6  =  -  5;  i.e., 
4-  divided  by  —  gives  — . 

Thus  the  rule  of  signs  in  multiplication  holds  in  division 
also;  viz..  like  signs  give  plus,  and  unlik(*.  minus. 


48  TEXT-BOOK    OF   ALGEBRA. 

44.  Exercise  in  the  Division  of  Algebraic  Numbers. 

I.  Divide  +  10  by  -  2.        2.    Divide  —  15  by  3. 
3.   Divide  -  50  by  5.  4.   ^  ^  — 6  =  ? 

5.    _3.i  ^4?  6.    1.06 -^  -9? 

7.  _  256  -J-  -  8000  ?  8.    ~  ^^'^^ 

—  2^ 

9.  ^    X  -^  X  -  ??  X  -  .007  -f.  -  -?  =  ? 
—  3         —  13  2  .3 

10.  If  in  11  days  the  mercury  falls  from  15°  above  zero 
to  18°  below,  how  much  is  that  a  day,  and  how  shall  we 
mark  the  terms  of  the  division  ? 

II.  A  man  losing  5  cents  on  every  bushel  of  wheat  he 
sold  found  he  had  fallen  short  $400.  How  many  bushels 
did  he  sell  ?     Mark  the  terms. 

45.  Roots  of  Opposite  Numbers.  —  The  term  root  of  a 
number  is  used  in  the  same  sense  in  algebra  as  in  arith- 
metic. 

Thus,     V^  =  0;    ^-64  =  -  4,  etc. 

a.  The  radical  sign  ( y/  )  is  used  to  show  that  a  root  is  to  be 
taken.  What  root  is  desired  is  indicated  by  the  small  figure  above 
and  to  the  left.  When  no  figure  is  written  the  square  root  is  under- 
stood. 

It  will  be  convenient  to  first  take  up  square  roots,  then 
cube  roots,  and  lastly  other  roots.     As  in  Powers,  the  si^ns 
of  the  roots  is  the  main  point  considered. 
1.    8(piare  Hoots. 

(1.    The   s(piare   root  of   a  positive   number  maij  be 
cither  pins  or  minus. 

Thus,     V  4  =  +  2  or  -  2.       For,      +2x+2=+4; 

_also  -2x-2=+4. 
V  28  =  -h  5.29+,  or  -  5.29+.     For,      +  5.29  X  +  5.29  = 
+  28  ;    also  -  5.29  x  -  5.29  =  +  28 ; 
and  so  for  any  other  number. 


ALGEBKAIC   NOTATION.  40 

b.  To  mark  the  two  roots  without  rewriting  the  number  the 
sign  ±  (read  "plus  or  minus")  is  used.  It  indicates  two  different 
numbers. 

(2.  The  square  root  of  a  negative  number  is  impos- 
sible. For,  the  square  root  of  any  negative  number,  say 
of  —  36,  to  be  an  algebraic  number  must  be  either  -|-  6  or  — 
6.  Xow  it  is  7ieither,  since  (-)-  6)-  =  +  36,  and  (—  6y  = 
-{-  36  also.     Hence  —  36  has  no  algebraic  square  root. 

2.  Cube  Roots. 

(1.    The  cube  root  of  a  positive  number  is  i)ositive. 

Thus,     ^/Tt  =  +  3,  since  (-}-  3)»  =  +  27. 

(2.   The  cube  root  of  a  negative  number  is  negative. 
Thus,     V-~Gi  =  -  4,   since  (-  4f  =  -  64. 

3.  Other  Roots. 

(1.  Roots  whose  index  is  a  prime  number.  —  Of  these, 
positive  numbers  have  a  positive  root,  and  negative  numbere 
a  negative  root,  as  in  the  cube  root. 

(2.  Roots  whose  index  is  a  composite  number.  — 
Such  roots  may  be  found  by  extracting  the  roots  denoted 
by  the  factors  of  the  index,  the  one  after  the  other.  Thus, 
the  fourth  root  can  be  derived  by  extracting  the  square 
root  twice,  the  sixth  root  by  extracting  the  s(juare  root  first 
and  then  the  cul)e  root  of  the  result.  The  fifteenth,  by  ex- 
tracting the  cube  root  and  the  fifth  root,  and  so  on. 

4.  Rules  for  signs  of  rcxjts.  —  We  saw  above  that  the 
square  root  of  negative  numbers  is  impossible.  Hence  we 
have 

(1.    Even  roots  of  negative  numbers  are  im[)ossible. 

(2.  Even  roots  of  jiositive  numbers  are  to  be  writ- 
ten ^. 

(3.  Odd  roots  of  numbers  have  the  same  sign  as  the 
powers  themselves. 

Note.  — These  rules  have  reference  only  to  wliat  are  called  reed 
roots.     This  subject  will  be  treated  more  fully  in  Chai)(er  XVIH. 


50  '      TEXT-BOOK   OF   ALGEBRA. 

46.   Exercise  in  extracting  roots  of  algebraic  numbers. 

1.  Extract  the  square  root  of  36,  of  109,  of  49,  of  225,^ 
and  of  121. 

2.  Extract  the  cube  root  of  8,  of  —  27,  of  —  64,  of  125, 
and  of  -  343. 

3.  Extract  the  fourth  root  (i.e.,  the  square  root  twice)  of 
16,  81,  and  of  256. 

4.  Extract  the  sixth  root  of  64,  of  729. 

5.  What  algebraic  numbers  are  those  whose  squares  are 
each  +  144  ? 

6.  What  algebraic  number  is  it  whose  cube  is  —  512  ? 

7.  W^hat  is  the  length  of  one  side  of  a  cubical  box  whose 
content  is  216  cubic  inches  ? 

1  If  the  student  is  not  farniliiir  with  tlie  rules  for  finding  square  and  cube 
roots,  the  numerical  results  will  have  to  be  found  by  trial  multiplications. 


ALdKIJKAIC    NorATloN.  51 


CHAPTER    III. 

LETTERS   USED  TO    KEPKESENT   NUMBERS,  DEFINITIONS, 

A  N  L)    K  X  P  L A  N  AT  1  ON  S. 

47.  The  Use  of  Letters  to  Stand  for  Numbers  constitutes  the 
seciond  distinguishing  chanu^teristic  of  algebra.  We  pro- 
ceed to  give  some  ilhistrations  of  this  use  l)efore  taking  up 
the  next  topic.  The  solutions  of  the  following  questions 
show  how  a  letter  (x)  may  represent  different  numbers  in 
different  problems. 

1.  What  number  Is  that  which  being  added  to  twice  itself  the 
sum  is  42  ? 

Solution.  —  Let  x  =  the  number, 

then  2x  =  twice  the  number,  (2a; means  2  times  x) 
and  X  +  2  x  =  the  sum. 
But  the  sum  is  42. 

Tlierefore  x  +  2  x  =  42, 

or,  8  X  =  42  (for  x  +  2  x  =  3  x). 

Now  if  3  X  =  42,  1  X  or  X  =  i  of  42  =  14. 
Therefore  x  =  14,  which  is  the  number. 

2.  What  number  is  that  to  which  if  we  add  its  half  and  ten  more 
the  sum  is  43  ? 

Let X  =  the  number  ;  then  -  x  =  one-half  the  number,  and  -x  = 
2  2 

the  number  +  -  of  the  number. 
2 

Hut  the  iuiml>er  +  -  the  number  -H  10  =  43. 
•> 

Therefore  -x+  10  =  43. 


52  TEXT-BOOK    OF    ALGER  11  A. 


Now,  if  10  added  to  ^  ic  =  43,  |  x  =  43  less  10  =  33; 

If  -  X-  33,  then  -  x  =  i  of  33  =  11: 
2  '  2  3 

and  —X,  i.e.,  x  =  2  x  11  =  22. 

2     ' 

3.  In  a  store-room  containing  40  barrels  the  number  of  those 
that  are  filled  exceeds  the  number  that  are  empty  by  16.  How  many 
are  there  of  each  ? 

Let  X  equal  the  number  filled,  then  x  —  16  equals  the  number 
empty. 

Hence,  x  +  x  —  16  =  40, 

or,  2  X  -  16  =  40. 

Now,  if  16  has  to  be  subtracted  from  2  x  to  give  40,  then 

2  X  =  40  +  16  =  56 

X  =  28  barrels  filled, 

X  —  16  =  12  barrels  empty. 

4.  Three  pieces  of  lead  together  weigh  47  lbs. ;  the  second  is 
twice  the  weight  of  the  first,  and  the  third  weighs  7  lbs.  more  than 
the  second ;  what  is  the  weight  of  each  piece  ? 

Let  X  =  the  number  of  lbs.  the  first  piece  Aveighs  ; 
then     2  X  =  the  number  of  lbs.  the  second  piece  weighs, 
and       2x4-7  =  the  number  of  lbs.  the  third  piece  weighs. 

But  the  sum  of  these  weights  is  47  lbs. ; 
therefore,       x  +  2x-f-2x-|-7  =  47, 
or,  5  X  +  7  =  47. 

Now,  if  7  added  to  5  x  is  47,  5  x  =  40.     Hence, 

x  =  8 
2x  =  16 
2  X  +  7  =  23 

5.  A  boy  bought  a  certain  number  of  lemons  and  twice  as  many 

oranges  for  10  cts.,  the  lemons  costing  2,  and  the  oranges  3  cts.  apiece; 
how  many  were  there  of  each  ? 

Let  X  =  the  number  of  lemons, 


AI.r.KrULMC    NOTATION.  58 

Now,  X  lemons  at  2  cts.  apiece  amount  to  2  x  rts.,  and  2  x 
oranges  at  3  cts.  apiece  amount  to  (>  x  cts.  Then,  to  find  the  cost  of 
both,  2x  +  6a;  =  40. 

8  X  =  40. 

X  =  5 

2  X  =  10 

6.  A  father  gave  his  boy  three  times  as  many  cents  as  he  had, 
his  uncle  then  gave  him  40  cts.,  when  he  found  he  had  nine  times 
as  many  as  he  had  at  first  ?     How  many  had  he  at  first  ? 

Let  X  =  the  number  of  cents  he  had  at  first,  then  3  x  =  tlie  num- 
ber of  cents  his  father  gave  him,  so  that  4x  =  the  number  he  now 
had. 

His  uncle  gave  him  40  cts.  more,  when  the  statement  is  made 
that  9x  =  his  total  amount. 

Hence,  4  x  +  40  =  1)  x. 

It  is  plain  now,  that  if  40  added  to  4  x  makes  9«,  then  40  =  5  x. 

If  5  X  =  40, 

X  =  8  the  number  he  had  at  first. 

Exercises  like  the  precetling  will  be  found  in  Art.  85. 


SECTIOX   I. 

Lki  1  hi;>   u  mi    ihk  Si<.n>. 

48.  An  Algebraic  Expression  is  anytliing  written  in  the 
algebraic  notation. 

Thus,  a,  -oO,  «2  ^  ah,  and  ^^^-^^^  are  all  alge- 
braic expressions. 

49.  As  suggested  by  the  expressions  just  written,  and  by 
others  already  given,  the  signs  appearing  in  arithmetic  are 
used  in  algebra  also,  and  retain  the  same  meaning.  Of 
them  the  most  impoi-tant  are 

-f-,— ,  x.-T-,(    ),=.  and  the  signs  of  powers  and  roots. 


54  TEXT-BOOK  OF  algp:bra. 

50.  The  Sign  +  (read  "  plus  "  )  denotes  addition,  and  is 
used  to  show  that  the  numbers  ^  between  which  it  is  placed 
are  to  be  added. 

Thus,  a  -{-  b  means  the  sum  of  the  numbers  denoted  by 
a  and  b. 

51.  The  Sign  —  (read  "  minus  "  )  denotes  subtraction,  and 
is  used  to  show  that  the  number  following  the  sign  is  to  be 
subtracted  from  the  other. 

Thus,  c  —  d  means  the  number  denoted  by  c  less  the 
number  denoted  by  d. 

52.  The  Sign  X  (read  "  multiplied  by  "  )  is  used  to  show 
that  the  numbers  between  which  it  is  placed  are  to  be 
multiplied. 

Thus,  a  X  c  means  the  product  of  the  numbers  denoted 
by  a  and  r. 

a.  The  sign  "  • "  is  sometimes  used  instead  of  X  to  denote  multi- 
plication. Thus,  6  •  b  '  c  means  that  the  numbers  denoted  by  5,  6, 
and  c  are  to  be  multiplied  together. 

b.  In  algebra  no  sign  indicates  multiplication. 

Thus,  loabc  denotes  the  product  of  the  numbers  15,  a,  6,  and  c. 


1  Letters  and  Figures  Representing  Numbers. —  A  word  Is  the  sign  of  :in 
idea.  Now,  a  word  (or  figure  or  letter)  that  stands  for  a  number  is  as  different 
from  the  number  itself  as  the  written  word  wagon  is  different  from  a  real  wagon. 
A  number  as  defined  in  arithmetic  is  a  unit  or  collection  of  units,  and  5,  or  five, 
for  example,  is  nothing  but  a  symbol  which  refers  to  this  particular  number. 
In  ordinary  language  we  drop  this  distinction.  To  illustrate :  an  older  person 
pointing  to  the  picture  of  a  horse  in  a  book  asks  "  What  is  this  ?  "  And  the  child 
immediately  replies  "  It  is  a  horse."  Neither  of  the  two  refers  to  its  being  only 
the  picture  of  a  horse.  So  to  the  class  in  arithmetic  the  teacher  says,  "  Tut 
down  the  number  fifty,"  when  a  more  correct  statement  would  be  "  I'ut  down  the 
figures  which  denote  fifty.  Such  ellipses  are  very  frequent  in  all  discourse. 
Since  much  is  gained  in  clearness  and  brevity  and  nothing  is  lost  if  this  explana- 
tion be  remembered,  it  seems  best  to  use  the  abridged  expressions.  Thus  in  the 
above  a  correct  statement  would  be  "the  sign  +  denotes  addition,  and  is  used  to 
show  that  the  numbers  between  whose  symbols  it  is  placed  are  to  be  added.  'J'he 
difference  between  numbers  and  symbols  of  numbers,  it  is  hoped,  is  here  plainly 
emphasized,  but  circuitous  language  in  definitions  to  insist  on  this  distinction  is 
avoided. 


ALGKHKAIC   NOTATION  65 

However,  it  must  be  understood  that  tliis  does  not  apply  to  the 
arable  symbols.  Thus,  2524  means  two  thousand  live  hundred 
twenty-four  in  algebra  as  well  as  in  arithmetic. 

53.  The  Sign  -r-  (read  '•  divided  by")  denotes  that  the 
iiiunber  preceding  the  sign  is  to  be  divided  by  the  number 
following  it. 

Thus,     Gab  -i-  -i  c  means  that  6 ab  is  to  be  divided  by  4 c. 

a.   The  fractional  form  ^  is  also  used  to  denote  a  divided  by  b. 
b 

54.  The  Parentheses,  ( ),  are  used  to  show  that  all  inside  is 
looked  on  as  one  number.     Thus,    (  --"J j  would  be 

.34 

treated  just  as  if  it  were  a  simple  fraction  ttt.  .     In  the  same 

way  f  '^  ~  '^^  ~^  o)  would  also  be  regarded  as  one  number. 

55.  The  Sign  =  (read  "  equals  to  ")  is  used  to  indicate 
that  the  algebraic  expression  on  its  left  has  the  same  numer- 
ical value  as  the  expression  on  its  right. 

a.  To  explain  further:  The  sign  =  denotes  that  the  numbers  or 
combinations  of  nrmbers  on  its  two  sides  reduce,  when  simplified,  to 
the  same  number.     We  proceed  to  illustrate  this:  — 

«-r2  +  4        11-2  X. 5 


7  X  :j  X  4  ~  3  +  18  -r  2  • 

For,  -— — — —  reduces  to ,  which  reduces  to  A; 

7X3X4  84    *  "' 

and  -^ '-  reduces  to  — — ,  which  reduces  to  ^; 

3+18^2  12  ^^ 

that  is,  when  reduced,  both  have  the  same  value,  and  therefore  they 
are  equal. 

Likewise,  ab  =  He  -\-  a  when  a  =  8,  6  =  C,  and  c  =  ,5. 
For,  ab,  or  8  X  6  is  48;  and  8  c  +  a  =  8  x  .5  +  8  is  48  also. 
Hence,  ah  =  S  c  -h  a,  when  a  =  S,  h  =  0,  r  =  5. 


Ef6  TEXT-BOOK   OF   ALGEBRA. 

56.  An  Exponent  ^  is  a  small  symbol  of  number  written  to 
the  right  and  above  another  number.  When  a  whole  num- 
ber, it  shows  how  many  times  the  other  is  used  as  a  factor. 

Thus,    5  X  5  —  ()-',  a  X  a  X  a  X  a  =  a* ',  2  a  X  2  a  X  2  a 
=  (2  ay.     ¥  means  c  ^'s  multiplied  together. 
If  c  ==  6,  then  ¥  =  h  X  h  X  h  X  h  X  h  X  h. 
a.   When  no  exponent  is  written,  1  is  understood :  — 

Thus,  a,  i.  e.,  a  used  once  equals  a'. 
6.    The  meaning  and  treatment  of  fractional  exponents  it  is  pre- 
ferable to  give  further  on,  in  Chapter  XIX. 

57.  The  Product  arising  from  taking  a  number  as  a  factor 
a  certain  number  of  times  is  called  a  Power  of  the  Number. 

Thus,     a  X  a  =  a^  is  the  second  power,  or  square  of  a. 
5x5x5  =  5^   is   the    third    power,,  or   cube   of    5. 

(a  -i-b)  X  (a  -\-  b)  X  (a  -{- b)  X  (a  -\- b)  =  (a  -\-  by   is 
the  fourth  power  of  a  -\-  b. 

a.  The  exponent  is  called  the  index  of  the  power. 

b.  The  second  power  of  a  number  is  usually  styled  its  square ; . 
and  the  third  power  of  a  number,  its  cube. 

58.  The  Sign  ^  (called  a  radical  sign)  is  used  to  indi- 
cate a  factor  which  multiplied  by  itself  some  number  of 
times   will  produce  the  number  under  the  sign. 

59.  An  Index  of  a  Radical  Sign  is  a  small  figure  or  letter 
placed  to  the  left  and  above  it,  to  show  into  how  many  equal 
factors  the  number  is  to  be  separated. 

Thus^    V25  =  i  5  ;  ^/64  =  4 ;  a/81  =  i  3  ;  -^32  =  2  ; 

Vi  =  1,  in  which  a  can  have  any  value  j  for  any  root 

of  1  is  1. 

a.    When  no  index  is  written,  2  is  understood.     Thus  VlO  =  ±  4; 

Va^  +  b^  means  a  number,  which,  multiplied  by  itself,  equals  a-  +  //-. 

1  From  Latin  ex,  "  out  of,"  and  po7w,  "  to  place;  "  i.  e.,  it  is  placed  out  from 
the  number  to  which  it  belongs. 


ALGEBRAIC  NOTATION.  57 

60.  One  of  its  equal  factors  is  called  a  Root  of  a  Number. 

Thus,     30  =  6  X  6,  therefore  0  is  a  root  of  36 ;  243  =  3 
X  3  X  3  X  3  X  3,  therefore  3  is  the  fifth  root  of  243. 

a.  One  of  two  equal  factors  is  called  a  square  root^  and  one  of 
three  equal  factors  is  called  a  cube  root. 

Thus,  4  a-  =  2  a  X  2a,  and  2  a  is  tne  square  root  of  4  a^.  —  6  is 
one  of  the  square  roots  of  oO;  y/Wl^  —  ±Jx,  read,  "the  square 
root  of  49 X-  equals  plus  or  minus  7x."  \/27  =  3,  read,  "the  cube 
root  of  27  equals  3." 

SECTION  II. 
Classification  of  Symijols. 

61.  A  Symbol,  as  used  in  Algebra,  is  a  letter  or  sign  with 
a  distinct  meaning. 

Thus,  a  letter,  as  a,  stands  for  a  numl)er ;  -|-  stands  for 
the  operation  of  addition;  and  "(  )"  indicates  that  an 
expression  inside  is  to  be  looked  u])()n  as  one  number. 

a.  The  Symbols  used  in  Algebra  are  (1)  S>nnbols  of  Number, 
(2)  Symbols  of  Operation,  (3)  Symbols  of  Relation,  (4)  Symbols  of 
Aggregation,  (5)  Symbols  of  Omission,  (6)  Logical  Symbols. 

62.  Symbols  of  Number.  —  Besides  the  use  of  letters  and 
figures  to  stand  for  numbers,  two  other  characters  are  used, 
0  (see  19),  and  oo . 

a.  The  sign  0  (read  "nought"  or  "zero")  either  denotes  zero, 
as  in  Arithmetic,  or  an  infinitesimally  small  number.  It  marks  the 
bejxinning  point  for  the  two  series. 

h.  The  sign  x  (read  "infinity")  stands  for  an  exceedingly  large 
number,  greater  than  any  that  can  be  named. 

63.  Symbols  of  Operation.  —  Tlie.se  are  -}-,  —^  X,  'y  -h, 
the  exponent,  and  the  radical  sign.  Division  is  also  indi- 
cated by  writing  the  dividend  over  the  divisor. 


58  TEXT-UOOlv    OF    ALGEBRA. 

64.  Symbols  of  Relation.  —  These  are  =,  >  <j  «:,  called, 
respectively,  the  signs  of  equality,  inequality,  and  variation. 

a.  A  horizontal  V,  >,  or  <<  is  used  to  show  that  the  numbers 
between  which  it  is  placed  are  unequal.  The  opening  is  always 
turned  toward  the  larger  number.  >  is  read  "greater  than;"  -<  is 
read  "  less  than." 

Thus,     8<7;  8  +  4>3  X  3;  G<10. 

h.  The  symbol  cr  means  "  varies  as."  As  the  number  on  its  right 
increases,  the  number  on  its  left  increases,  and  as  the  number  on  its 
right  decreases,  the  number  on  its  left  decreases  in  the  same  ratio. 

Thus,  a  man's  monthly  wages  a  the  number  of  days  he  works. 

c.  A  line  drawn  across  any  of  these  signs,  thus,  7^,  or  -^  is  used 
to  deny  that  which  the  symbol  unmarked  expresses  ;  e.  g.,  0  ^i  5, 
11  5^110. 

65.  Symbols  of  Aggregation.  —  These   are    (  ),    \    I,   [  ], 

,  I  .  Named  in  order,  they  are  parenthesis,  brace,  bracket, 
vinculum,  and  bar.  They  are  all  used  for  the  same  purpose; 
viz.,  to  show  that  an  included  expression  is  to  be  looked 
upon  as  one  number.     See    54. 

66.  Symbols  of  Omission.  —  Dots  or  dashes,  usually  called 
symbols  of  continuation,  are  used  to  show  that  certain  ex- 
pressions have  not  been  written,  and  are  to  be  filled  out 
from  those  that  are  given. 

Thus,  a,  a"^,  a^  .  .  .  a^^ ;  or,  x  -{-  x^  -■{-  x^  -\-  x*  .  .  .  x-'^' 

They  may  be  read  "  and  so  on  to." 

67.  Logical  Symbols.  —  The  logical  symbols  are  those  of 
reason  and  conclusion  The  sign  •.•  (read  "since"  or  ''be- 
cause") is  used  to  mark  a  reason.  The  sign  .-.  (read  "there- 
fore," "  hence  ")  is  used  to  mark  an  inference  or  conclusion. 


AH.KI'.KAIC    .N(»lAri(»N.  59 

SEf'TloN    III. 
Classification  <>i-  Ai.«.kiu: ak    Kxthessions. 

68.  A  Term  is  iiii  algebriiic,  expression  the  parts  of  which 
iiif  nut  separated  by  plus  and  minus  signs; 

Thus,  3  ab,  9  a%e,  10  x  are  terms.  Also  Aah  -\-2  ac  — 
,3  be  is  an  ex])ressiou  consisting  of  three  terms  :  the  first 
4  abj  the  second  2  ac,  and  the  third  3  be 

a.  A  simple  term  is  a  single  expression,  as  2  ax,  5  w,  ^p-  q. 

b.  A  compound  tenu  is  any  combination  of  simple  terms  looked 
upon  as  one  expression.     Tliis  is  usually  indicated  by  a  symbol  of 

(cc        c 

69.  An  algebraic,  expression  consisting  of  but  one  term 
is  called  a  monomial. 

Thus,  Cyabr,  11  ?;/-,  16  ax  are  each  monomials. 

a.  It  is  plain  that  both  wortls  monomial  and  term  can  ])e  applied 
to  the  same  expression  and  with  the  same  intent. 

To  illustrate:  3  a6c  can  be  called  either  a  term  or  a  monomial. 
Tenu  is  the  more  general  word  of  the  two,  and  may  cover  compound 
expressions.  Monomial  is  more  explicit  in  referring  to  a  single 
tcrni. 

70.  A  Binomial  is  nn  algebraic  expression  consisting  of 
two  terms. 

Thus,  II  -f-  b.     'J  11  —  '•.     '/-//  —  f'-. 

71.  A  Trinomial  is  an  algebraic  expression  consisting  of 
thre.'  terms. 

As.  "  -\-  b  -{-  (',      (lb  -\-  be  -j-  tn\       'J  <ibr  —  ab'  -\-  br'-. 

72.  A  Polynomial  is  an  algebraic  expression  consisting  of 
many  terms,  usually  taken,  however,  to  mean  two  or  moie. 
Thus.  „  -\-  b  -\-  c  i-  d  ;  a^  -  b'  -  r^  -  </*  -  e^ ;  3  mn  -  4  m^ 

-I-  r>;/2-j-  10  m*;/^ 


60  TEXT-BOOK    OF   ALGEI'.RA. 

SECTION    lY. 
On  the  Term. 

73.  A  simple  term  in  algebra,  such  as  5  a^  x,  usually  con- 
sists of  a  coefficient  and  a  literal  part,  each  letter  having 
an  exponent  expressed  or  understood. 

74.  The  Coefficient  ^  shows  how  many  times  the  number 
denoted  by  the  product  of  the  other  factors  is  taken. 

In  5  a^x,  (i.  e.,  a'^x  -f  a^x  +  a'^x  +  a^x  -}-  a^x),  5  is  the 
coefficient,  and  shows  how  many  times  the  number  denoted 
by  a:^x  is  taken.  In  0  ax,  6  a  may  be  regarded  as  the  coef- 
ficient, showing  how  many  times  x  is  taken.  Of  course, 
the  a  is  a  literal  factor ;  but  we  may,  if  we  choose,  regard 
some  of  the  literal  factors  as  part  of  the  coefficient.  In 
3  {a  +  h),  [i.  e.,  {a  +  ^)  +  («  +  h)  +  (^  +  ^)],  3  is  the 
coefficient  of  {((  -\-  h). 

a.  The  coefficient  standing  in  front  of  its  literal  part  shows  how 
many  times  that  part  is  to  be  added;  while  the  exponent  of  an 
expression,  written  to  the  right  and  above  it,  shows  how  many  times 
it  is  to  be  used  as  a  factor  in  a  nudtiplication. 

To  illustrate-:  5  ah  means  ah  +  ah  +  ah  +  ah  +  ah  \  while  {ah)^ 
=  ah  X  ah  X  ah  X  ah  X  ah. 

h.  The  coefficient  1  is  not  written,  although,  of  course,  it  may 
be  if  desired.     Thus,  he  and  1  he  are  the  same  thing. 

75.  The  Literal  Part  of  a  term  is  the  part  containing  the 
letters. 

In  10  a%,  a%  is  the  literal  part. 

a.  One  letter,  as  a  in  a%,  is  sometimes  called  a  dimension  of  the 
term. 

1  From  Latin  co  or  con,  "with,"  and  efficio,  "to  effect,  or  make;"  i.e.,  the 
coefficient  ii  tliat  which  with  the  literal  part  makes  up  the  term. 


ALGEBRAIC   NOTATION.  61 

76.  Similar  Terms  or  Like  Terms  are  those  whieli  have  the 
same  letters  affected  by  the  same  exponents. 

7  aU^  and  9  cU/^  are  similar  terms.  So  also  ^re  11  x^y^  and 
14  ^y.  Again,  ax'^z  and  mx'^z  are  similar  with  respect  to 
the  literal  i)art  x'z. 

a.  In  arithmetic  we  could  add  and  subtract  like  numbers,  as 
$7  —  $5  =  $2  ;  19  qts.  +  10  qts.  =  35  qts.  Likewise  we  can  add 
8  times  a  certain  number  to  5  times  the  same  number  and  get  13 
times  the  number.  In  the  same  maimer  we  can  add  and  subtract 
what  have  been  defined  as  similar  terms. 

Thus,  1  ab^ -\- \)  ab'- =  Uiab-. 
For,  if  a  =  4  and  6  =  3,  ab-  =  4  X  3-  =  30. 
And  7  X  30  +  9  X  30  =  10  X  30. 
Likewise,  11  x^  y^  -  8  x*»  y^  =  3  r'  y^. 

b.  For  like  and  unlike  signs  of  tenns,  see  26. 

77.  The  Degree  of  a  Term  is  the  sum  of  the  exponents  of 
its  literal  factors.     (Remember  56,  a.) 

The  expression  a%c  is  of  the  fourth  degree. 
6  mhi  is  of  the  third  degree. 
9  x^i/*z*  is  of  the  ninth  degree. 

78.  Homogeneous  Terms  are  those  which  are  of  the  same 
degree.     The  following  sets  are  homogeneous  : 

6  a%c^f  7  ab^c%  9  a%^c  all  of  the  fifth  degree. 
9  ary,  11  a:«,  4  x*x^,  yz^  all  of  the  sixth  degree. 

a.  A  polynomial  is  said  to  Ik*  liomogeneous  when  all  its  tonus  arc 
of  the  same  degree. 

To  illustrate:  (>  ni^n^  —  7)  m^n  ;/'  —  ;>-'y»  —  12  m'^-q-  is  homogeneous. 
So  also  is  cC'b  —  ab-  —  abc  —  a:-c  +  ac-  +  be- 


02  TEXT-BOOK    OF   ALGEBRA. 


CHAPTER  IV. 

EXERCISES    IN   THE   NOTATION. 

SECTION    I. 

Exercise  in  Reading  Algei3ijaic  Expressions. 

79.   Read  the  following  expressions  : 

1.    a  -^  b  —  c.  2.    X  -\-  X  1/  —  I/. 

Remark. — If  the  first  were  to  be  read  as  suggested  in  50  and  51 
we  should  say  "  the  sum  of  the  numbers  denoted  by  a  and  6,  dimin- 
ished by  tlie  number  denoted  bye.'"  The  second  should  read  "the 
sum  of  the  numbers  denoted  by  x,  and  the  product  of  the  numbers 
denoted  by  x  and  y,  less  the  number  denoted  by  y."  It  is  assumed, 
however,  that  by  this  time  the  student  is  familiar  with  the  idea  of 
a  letter  standing  for  a  number,  and  it  will  be  sufficient  to  read 
the  first  "a  plus  6  minus  c,"  and  the  second  "x  plusx  times  y 
minus  ?/." 

3.  V2  a  -  be  +  d  (52,  b).  5.    a''  +  3  «.  ( 57 ). 

4.  2  a  +3  b  -  c.  6.    2  d'  -\-Ab^-  3  cl 

7.    tt  +  Z/  -f  r  +  (/. 

Kk.makk.  —  This  may  be  read  "«  plus  6,  plus  r,  plus  d  ;"  but  it 
is  regarded  as  more  elegant  to  say  "the  sum  of  «,  6,  c,  and  fZ." 
Also  to  use  this  mode  of  expression  in  reading  binomials  particularly. 
Thus,  ex.  5  would  be  read,  "  the  sum  of  a  squared,  and  three  times 
a."'  Likewise,  a  residual  binomial,  such  as  a-  —  4  6-  would  be  read 
"  the  difference  of  a  squared  and  four  times  h  squared. 

8.  (;  ab  +  9  ar  +  14  br.      10.    4  .r-  —  \)  //-. 

9.  44  abc  —  11  bed.  11.    24  a  -  5  A  +  14  c  -  7  d. 


AL(;i:iiKAIC   NOTATK^N.  63 

12.  8  abc  —  bed  -I-  i)  cde  —  def. 

13.  V"  H-^  (59,  a). 

14.  c*  H-  rf2,  or  ^  (53,  a). 

15.  7  «'  -  3  «'-^/»  +  c». 


SECTION  II. 

K\KK(  isK  IN  AVkiiim;   Al(jkj;j:ak    Kximjkssions. 

80.  Kxercise  in  writing  expressions :  — 

1.  Express  the  sum  of  a,  6,  and  c. 

2.  Express  the  double  of  h. 

3.  Express  a,  plus  h  divided  by  c. 

4.  By  how  much  is  a  greater  than  5  ? 

5.  Write  the  sum  of  a  cubed,  thn»e  times  c  siiuaicd,  and 
the  product  of  />,  r,  and  d. 

6.  Write  cd  over  i,  i)lus  four  times  b  divided  by  three 
times  a^  minus  cd  divided  by  24. 

7.  Write  the  sum  of  b  .scpuire^l  and  c  scpiared,  divided  by 
the  (UfTerenee  of  two  times  e  and  three  times  a. 

8.  Express  (•  to  the  fourtli  ])Ower.  less  four  times  /•(•u!»ed. 
plus  three  times  c  less  G. 

9.  Write  tlie  sum  of  the  sixtli  jiowers  of  <i  and  A. 

10.    W'lite  tin-  (btferenee  of  the  tifth  powers  of  a:  and  //. 


64  TEXT-BOOK   OF   ALGEBKA. 

SECTION   III. 
NuxMEKicAi.  Values. 

81.  The  Numerical  Value  of  an  algebraic  expression  is 
the  number  obtained  by  giving  a  particular  value  to  each 
letter  and  then  performing  the  operations  indicated. 

82.  Evaluate  the  following  expressions  : 

1.  c  -\-  d  —  b  when  c  =  5,  d  =  10,  b  =  3. 

2.  6  a-  ic  —  9  ?/^  when  a  =  A,  x  =  2,  ij  =  3. 

3.  3a^  -{-2  ex  —  b^  when  a  =  5,  b  =7,  c  =  15,  x  =  3. 

4.  ^  +  ^  -^Aju.^  when  a  =  4,b  =  3,  c  =^  5,  d  =  10,  x  =  2. 


5.  -y/  a^  -{-  b'  -{-  c,  when  a  =  S),b  =  l,c=  14. 

6.  V«  +  V^  +  \/~c,  when  a  =  Si,  b  =  121,  c  =  64. 

7.  Obtain  the  numerical  values  of  the  expressions  in  80 
when  a  =  1,  &  =  2,  c  =  3,  tZ  =  4,  e  =  5,  /  =  6,  .r  =  10, 2/  =  5. 

83.  It  was  pointed  out  in  76,  a,  that  similar  terms  can  be 
united  into  a  single  term  by  combining  their  coefficients. 

Thus,  1  abG-{-3  abc  —  2abG  =  S  abc.  And  if  6*  =  2,  ^  =  3, 
c  =  1,  then  8  abc  =  48,  which  is  the  same  result  as  would 
be  obtained  by  substituting  in  each  term,  and  then  uniting 
the  separate  results.  7  abc  =  42 ;  3  abc  =  18  ;  2  abc  —  12, 
and  42  +  18  -  12  =  48. 

But  if  the  literal  parts  were  not  the  same,  they  would 
not  have  the  same  numerical  value,  and  could  not  be  added. 

Thus,  x'^z  and  xz^  are  not  similar,  and  so  we  find  when 
X  =  2  and  z  —  3,  that  the  first,  xH  =  12,  and  the  second, 
xz''=^  18. 

Again,  if  we  regard  part  of  the  literal  factors  as  belong- 
ing to  the  coefficients,  we  can  evaluate  certain  kinds  of 
exercises  more  readily. 


ALGKliilAil     NoTAlIoN.  .6/5 

Thus,    ax'-y  -J-  hxhj  —  3  cx^tj  -\-  4  dx'^n,  when   a  =  5,  b  = 
,S,  e  =  2f  d  =■  4,  and  x  =  S,  1/  =  7  becomes  (fr  -f  />  —  o  r  -f- 

4 r/) j-'V  =  (5+3-6  + 16)  X  :>-  X  7  -  IS  X  <;;;  =  1134.  J,is. 

1.  8 a:- 4- (> if- —  r>  J-- =  ?  when  ./•  —  1. 

2.  y x-//•^'  —  ^' *^'V'  +  ^^  *^'"//'  =  "•'  ^^I'^-ii  ^*  =  ->  y  =  ^^)  •"^♦i 

;i'  =  5. 

3.  a^x-z  4-  3  ^xv;  —  U  (//>^^^'  =  ?  "vvhen  a  =  o,  b  =  1\    and 
:r  =  3,  5;  =  -. 

4.  2  tn'tfz-  —  'J  n'-if-'S-  -|-  '*'intt/-.'r  =  '.'   \n  hrn  v/i  =4,  7t  =  3, 
and  y  =  1,  .^  =  -. 

84.  Numerical  Values  introduced  to  verify  Equations. 

Show  that  the  condition  of  an  (Mjuality  is  I'ullilhMl  in  llie 
following: 

1.  ^_~_^_  =  a  —  x,  when  a  =  it,  x  =  3;    wlicn   (t  =  (), 

a  -\-  x 

a;  =  L 

2.  ^^  —^  —  '^^  =  a;  —  0,  when  ^  =  S ;  also  when  x  =  1). 

a;  +  5 

3.  ?_ZL^  =  x^  -|-  J-//  -f  y-,  taking  x  =  any  nund)er,  y  = 

a;  -  y 

any  luimber. 

85.  To  find  the  Numerical  Value  of  a  letter  in  an  equation. 
Take  the  solutions  in  Article  47  as  models. 

1.  a:  +  6  =  20,  a;  =  ?     If  6  added  to  x  equals  20,  x  must 

equal  14. 

2.  2x4-6  =  20.  5.    3  ir  —  6  =  24. 

3.  2a:  -  4  =  20.  6.  .3x  +  I'O  =  5x  (.-.20;  =  10). 

4.  3  X  +  6  =  24.  7.    9  r  -  16  =  o  x. 

8.    Hx  —  35  =  x. 


QQ  TEXT-BOOK    OF    ALGEBRA. 

9.  John  is  three  times  as  old  as  James  and  the  sum  of 
their  ages  is  36 ;  how  old  is  each  ? 

Let  X  =  James'  age, 

then  3x  =  John's  age, 

and X  ■-}-  3x  =  the  sum  of  their  ages. 

But  the  sum  of  their  ages  is  3(j. 

.'.  X  -\-  3  X  =  3Q 
4.x  =36 

-     =^    \Aas. 
3x  =  21\ 

10.  There  are  four  times  as  many  girls  as  boys  in  a  party 
of  60.     How  many  were  boys  and  how  many  were  girls  ? 

Let  X  —  the  number  of  boys,  then  cc  +  4  cc  =  60  is  the 
equation. 

11.  Divide  a  line  21  inches  long  into  two  parts,  such  that 
one  part  may  be  |  of  the  other. 

SuGGESTiox.  X  +  3  X  =  21;  I  X  =  21,  J  ic  =  I  of  21  =  3;  |  x,  or 
x  =  4  X  3=  12:  21  -12  =  9. 

12.  A  stick  of  timber  40  feet  long  is  sawed  in  two,  so 
that  one  part  is  f  as  long  as  the  other.  Eequired  the 
length  of  each. 

13.  A  farmer  sold  a  sheep,  a  horse,  and  a  cow  for  ^105. 
Eor  the  horse  he  received  10  times,  and  for  the  cow  4  times, 
as  much  as  for  the  sheep.     How  much  did  he  get  for  each  ? 

14.  A  man  being  asked  how  much  he  gave  for  his  watch 
replied :  If  you  multiply  the  price  by  4,  to  the  product  add 
70,  and  from  the  sum  subtract  50,  the  remainder  will  be 
^20,     What  did  his  watch  cost  ? 


ALCF.IilJAlC    NOT  A  I  ION.  67 

15.  Three  boys  together  spend  50  ets.  :  the  second  spends 
5  cts.  more  than  the  first,  and  the  tliird  three  times  as 
much  as  the  first.     How  many  ( t  nis  did  each  spend '! 

16.  Says  A  to  B,  "  Good-mornmg,  master,  with  your  hun- 
dred geese."  Says  B,  "  I  have  not  one  hundred,  but  if  I 
had  twice  as  many  as  I  now  have  and  ten  more  I  should 
have  one  hundred."     How  many  geese  had  he  ? 

17.  An  apple,  a  ]>ear,  and  a  peach  cost  19  cts. :  a  pear 
costs  2  cents  more  than  an  apple  and  a  peach  3  cents  more 
than  a  pear.     How  much  did  each  cost  ? 

18.  Distribute  $3.50  among  Thomas,  Richard,  and  Henry, 
so  that  Richard  shall  have  twice  as  much  iis  Thomas,  and 
Henry  twice  as  much  as  Richard. 

19.  Divide  the  number  60  into  three  parts,  so  that  the 
second  may  be  three  times  the  first,  and  the  third  double 
the  second. 

20.  \Vhat  number  is  that  from  which  we  obtain  the  same 
result  whether  we  multiply  it  by  four  or  subtract  it  from 

400? 

21.  A  boy  bought  the  same  number  of  tops,  of  marbles, 
and  of  balls  for  65  cts.  :  for  the  marbles  he  gave  1  ct.  each, 
for  the  tops  2  cts.  each,  and  for  the  balls  10  cts.  each. 
How  many  of  each  did  he  buy? 

22.  <  )ut  of  77  lK)oks  a  certain  number  were  sold,  when 
tlnif  remained  three  more  than  were  sold?  How  many 
wnv  >nl,r/ 

23.  Jolin  bought  a  certain  number  of  tops  and  fifteen 
times  as  many  marbles.  After  losing  14  of  the  marbles 
and  giving  away  30,  he  had  only  16  left.  How  many  tops 
did  he  buy  ? 

24.  A  man  bought  3  liorses  and  4  cows  for  $600.  Each 
horse  cost  twice  as  much  as  a  cow.  How  much  did  lie  give 
for  eiich? 


68  TEXT-BOOK   OF   ALGEBRA. 

SiJGGESTiox.  —  Let  X  represent  the  price  of  a  cow,  and  2  x  that 
of  a  horse.     How  much  did  all  the  cows  cost,  and  all  the  horses? 

25.  Seven  men  and  three  boys  were  hired  for  a  week 
(6  days),  a  man  receiving  three  times  as  much  as  a  boy. 
Altogether  their  wages  amounted  to  $72.  What  did  each 
receive  ? 

Additional  problems  may  be  found  in  Tower's  Intellectual 
Alarebra. 


AI.ci.r.llAIC    KOTATION.  69 


CHAPTER   V. 

ADDITION. 

86.  Addition  ^  is  the  process  of  uniting  two  or  more  ex- 
pressions into  one  called  their  sum.  The  sum  is  usually 
obtained  in  its  simplest  form. 

a.  Meaning  of  Algebraic  Addition.  —  An  algebraic  expression  was 
defined  (48)  as  anything  written  in  algebraic  language.  In  Chapter 
II.  the  rules  for  adding  algebraic  numbers  were  given.  We  now 
proceed  to  apply  the  same  rules  for  signs  to  literal  expressions. 

87.  Examples  in  Addition.  —  It  is  convenient  to  classify 
examples  in  addition  into  four  cases  : 

1.  When  the  terras  added  are  similar  and  have  like 
signs. 

(1.   Monomials.     (See  28.) 

(1)     2  a  (2)  -      b    (3)  Sxy   (4)  -ISaV 

8  a  -10b  6xy  -    2  ah/ 

9  a  —  Ab  Oa-y  -20a^i/ 
12a  — _'?i  15  xY  —  aV 
;n  a  (See  76,  a.)  -  18  b  38  xY  -  41  a^ 

(2.    I  Polynomials.     (72.) 

(1)    Add  the  three  trinomials  given  below. 


4a^x- 
5  a^x  - 
2a^x- 

6 
14 

ah/  -\-    9  //2 

ah/-\-n)r' 

ah/  -\-10if 

n  a^x  -  21  ahj-\- 36  if 

'  The  etisential  deflnition  waa  given  in  Art.  16. 


70  TEXT-BOOlv   OF   ALGEBRA. 

This  solution  evidently  proceeds  on  the  assumption  that  the  three 
trinomials  can  be  added  part  at  a  time.  Thus,  we  first  add  the 
quantities  in  the  left  column,  next  those  in  the  middle  column,  and 
lastly  those  in  the  right  column.  Is  this  justifiable  by  the  nature  of 
addition  ?     See  Commutative  Law,  28. 

2.  When  the  terms  added  are  similar,  but  have  unlike 
signs. 

(1.  Monomials.  To  add  similar  terms,  add  their  coef- 
ficients (76,  a).  Consequently,  in  the  examples  now  to  be 
given  the  coefficients  are  added  by  the  rules  of  29,  and  the 
resulting  coefficient  is  prefixed  to  the  common  literal  part. 

(1)    —  6  a'^         Explanation.  —  The  sum  of  the  posi- 

+   Q       2 
f  %     ^^^'^  coefficients  is  11;  the  sum  of  the  nega- 

T  4  ^^2     tive  coefficients  is  9 ;  the  difference  is  2, 

_  3  fi'    ^iifl  the  sign  of  the  greater  is  +•     Hence, 

2^1^     +  2  is  the  coefficient  of  «^  in  the  sum.  . 

(2)            bxy  (4)            Snm^ 

—  Wdcy  —    2  71171^ 

—  14  xy  +21  m7i^ 
-{-    7  xy  —  72  )fi7i^ 

— xy  —  11  7)171^ 


—  14  xy 

(3)    ■        9xz'^  (5)         100  abc 

—  xz'  4  abe 

—  7  xz'  —  92  abc 
+  3xz^  +  42  abe 
+  10  xz""  -  9  abc 
-\-Uxz'- 

(2.    Polynomials. 

The   columns    are   added    separately   by   the     rule    for 
iiKmomials. 

(1)  a-2b-\-3c-4:d 

-2a  -{-3b  -4c  -i- 5  d         Remark.  —  The    third 

—  Aa  -{-  5  b  —  4-c  -{-  7  d     column  cancels  out  of  the 

3  a  —  4  i'  +  5  c  —  6  (/ 


sum. 


_  2 


2  a  +  2  /;  +  2  ^ 


ALGEBIIAIC   NOTATION.  71 

3.  When  the  terms  to  be  added  are  not  similar,  or  some 
similar  and  others  dissimilar. 

(1.  Add  8  a»x-  -  li  ax,  7  n.r  -  r»  .n,\  \)  .,•>,*  -  5  ax  +  2  x', 
and  2  a^x  ^  -\^  xtj* -\- H  if. 

Operation.      8  a^x-  —  3  ax 

7  ax  —  5  xy^ 
-  o  </x  +  9  J-//*  +  2  x^ 
2aV  -h     a;y^ -f  Sy* 

10  «V2  _~;7:r -h  5  0-?/*  +  2  x-^  -h  8  y* 

Explanation.  —  Similar  terms  are  placed  in  the  same  column, 
changing  the  order  in  the  expressions  if  necessary.  Thus,  the  —5  ax 
of  the  third  expression  is  written  down  to  the  left  of  the  9  xy*  which 
came  before  it,  as  given  in  the  problem.  Moreover,  as  fast  as  new 
terms  are  obtained,  they  are  written  in  new  columns  at  the  right. 

Remark.  —  In  connection  with  this  case  it  is  proper  to  say  that 
since  addition  is  merely  uniting  algebraic  expressions  to  obtain  the 
sum,  it  is  sufficient  in  all  cases  to  write  the  expressions  one  after 
another  with  their  proper  signs.  But  the  definition  says  the  sum  is 
usually  obtained  in  its  simplest  form.  The  processes  explained 
above  giv«*  the  results  in  their  simplest  forms  provided  that  polyno- 
mials which  have  similar  terms  are  first  simplified  as  explained  in  the 
next  ease. 

4.  A\'iuMi  tlie  tiM-nis  to  Ik*  added  constitiit*'  w  siiitifle  poly- 
nomial having  some  of  its  terms  similar. 

Simplification  of  ii  ixdynomial. 

(1.    Simplify      .'}  aV>  -f-  '.)  <i-r  —  7  a-h  -f-  4  aV  —  2  a^c 
+  4  a-b. 
:\  aV,  —  7a^b  +  A  u%  =  0  ;  9  a^c  —  2  a^c  =  7  a^c. 

Therefore  tlie  ]M)lynomial  reduces  to    7  a-r  -f  4  «V^ 

From  the  examples  aiul  explanations  here  given  we  derive 

the  rule  in  tin*  next  artich*. 


i'^  TEXT-BOOK    OF   ALGEBKA. 

88.  Rule  for  Algebraic  Addition. 

1.  Write  the  expressions  to  be  added  so  that  similar 
terms  (76),  shall  stand  in  the  same  column.  If  none  of  the 
terms  are  similar,  unite  them  just  as  they  stand,  case  3. 

2.  Add  the  coefficients  of  each  column  and  annex  the 
common  literal  part  to  the  sum.  In  adding  observe  the 
rule  in  29. 

3.  Any  terms  standing  alone  are  to  be  brought  down  with 
their  signs  unchanged. 

4.  To  simplify  a  polynomial  regard  each  term  as  one 
quantity,  and  proceed  to  add  sucli  as  are  similar  as  in  the 
first  part  of  this  rule. 

89.  Exercise  in  Addition. 

1.    Add  the  following  sets  of  monomials  : 

(1.  .*>  a,  5  a,  —  2  a,  7  a,  and  —  4  ((. 

(2.  10  am,  —  ()  (iin,  4  <f)n,  7  aiu,  —  9  am,  and  avi. 

(3.  -  3  dif,  4  dif  -  S  dif  -  13  dif,  2  d>f  and  18  dy^. 

(4.  —  6  a'^,  -\-  2  a-,  —  5  a'^,  4  a"-,  —  3  (r  and  a'-. 

(5.  13  vi^n,  —  10  ^n^n,  —  0  mhi,  5  mhi,  and  —  4  mhi. 

(6.  axy,  —7  axij,  -\-  8  axy,  —  axy,  —  8  axy,  and  +  9  axy. 

2.  Add  3  «^  +  o  a%,  6  ah  —  8  a'^h,  8  ah  —  a.%,  and  3  ah 
4-  2  an>. 

3.  Add  T)  ,r  -  3  a  +h  ^  7,  and  -  4  x  -  3  a  +  2  Z*  -  9. 

4.  Add  3  a  +  7  />  —  <S  r  -f-  d,  3  a  —  2  />  +  r  —  e,  and  —  a. 
_  h^  _  c,  -  d. 

5.  Add  7  .r-  —  2  x  —  5,  2  .^-^  —  3  a-  +  8,  and  —  9  if^  +  5 .7^ 
+  3. 

6.  Add  3  a%^  —  7  «^/^  +  T)  axy,  —  7  a.%^  —  2  ah*  —  axy, 
aJA  -  7  axy  +  8  aH}%  -  10  ah''  +  crb^  +  3  rtit-y,  and  -  5  a-^»^ 
+  18  ah\ 


ALdKBKAlC    NOTATION.  73 

Note.  —  Here  it  will  be  necessary  to  change  the  order  of  the 
tenns  as  in  87,  o.  In  the  review,  the  student  may  solve  these  prob- 
lems mentally  ;  i.  e.,  by  simply  running  the  eye  along,  picking  out 
similar  terms,  and  writing  down  th«'  sums  thus  obtained  with  their 
proper  signs.  By  this  plan  it  will  not  be  necessary  to  write  the 
problem  or  to  arrange  the  terms. 

7.  Add  2  (lb  —  lidx-  -|-  2  a-x,  Vl<ih  —  (I  a-,,-  -j-  fo  <,.r\ 
and  a  J**  —  8  «/;  —  i)  a'-b. 

8.  Add  .r^  -  2  a^  -h  3  x%  x^  -f  x-  +  .;•,  4  x'  +  ."»  x\  2  x'^ 

—  3  J-  —  4,  and  —  3  x-  —  2  x  —  5. 

9.  Siin]>lifv  .S  xij  —  o  a  -\-  (S  c  —  3  in  -f-  ."»  -f-  xij  -\-  V2 
-2  m. 

10.  Simplify  a  -{-  b  -\-  c  —  d  -- 2  e  —  2  a  -\-  ?y  b  —  r>  c 

—  r>  d  -\-a  —\b  -\-i\<l  —  3  <•  -f-  4  c  -f  5  c. 

11.  Simplify  8  a:»  -  1 ./  -  -f-  2  x^  —  5  a-  -  0  a--  +  a-^  -  2  -  or 

12.  Simplify  9 ^»  -3^4-15^—^6  +  ic-2d^A.c-p. 

13.  Add  2  J-"  +  3  3/*,  4  a-"  —  6  y*,  and  7  a;«  —  5  y''. 

14.  Add  10  irb  -  12  a^bc  -  15  iV  +  10,  8  a%c  -  10  6V 
_  4  a2^»  -  4,  20  bh"^  -  3  rt»ic  -  3  a'b  -  3,  and  2  a^^  + 
12  a«6c  +  5  ^»\*  +  2. 

15.  Add  a  4-  ^^  ^'  —  ^A  and  c  +  /  —  <j. 

16.  Add  5  nC-x-  -f  4  i5»«-^.r«  -h  mxhf,  and  10  caV  —  2  aZ^^x* 
-f-  G  mx'-if. 

17.  Of  two  farmers  the  first  had  2  «  —  3  y  acres,  and  the 
second  had  x  —  //  acres  more  than  the  first.  How  many 
iM3res  had  the  second  ?     How  many  acres  had  both  together  ? 

18.  One  man  is  worth  a  -\-  c  dollars,  a  second  is  worth 
n  -\r2b-\-  c  dollars,  a  third  is  worth  2  d  —  e  dollars,  while 
a  fourth  is  worth  a  -\- b  +/ dollars.  What  is  the  sum  of 
their  fortunes  ? 


74  TKXT-BOOK    OF   ALGEBRA. 


CHAPTER   VI. 

STMrril  ACTION. 

90.  Subtraction  is  the  ])r()cess  of  finding  from  two  given 
expressions  a  third  which  added  to  the  second  will  give  the 
first. 

The  two  given  expressions  are  called  respectively  minuend 
and  subtrahend,  and  the  third,  difference  or  remainder. 

91.  Examples  in  Subtraction. 
1,    ^Monomials. 

To  subtract  one  term  from  another  similar  to  it,  take  the 
difference  of  their  coefficients.     (See  31  and  32.) 

EXPLAXATION  OF  (2.  —  By 

31  and  32,  we  conceive  the 

(1.     tJ  (/  (2.     11  ir^     sign  of   the    subtrahend  co- 

*.)  (f'         ■         —  6  x'^     efficient  —  T)   to  be  changed 

>>  (0  17  x'^    to  +  (),  and  we  then  add  it 

to  11.     So  in  the  subsequent 

examples. 

(3. 


(4- 


2.    Polynomials.     (Will  it  be  correct  to  subtract  yw?'?^  fft  a 
time  just  as  we  added  ])art  at  a  time  iu  addition  ?) 


-  13  .ry^ 

10  Xff 

(5. 

(7. 

—  5  xy^ 
9  a;// 

—  2.'!  xff' 

—     (*.  (1 

-  ir>// 

(6. 

-  20  x' 

(8. 

-  12  ;/ 

-  ?>y' 

AL(JKlil:Ai<       N"iAili»N.  7") 

(1.    From     (>  a7>V  —  7  nhr^  -f-  1)  or  —  :\  hr*    take     4  it^b^r 

i  )j>e ration. 

6  n-O^r  —    7  (i/tt-^  -f-    *.)  ae  —    3  be*  minuend. 
4  a'-lr^c  -\-    8  dbc'^  —    S  ac  —  10  ^6'*  subtrahend. 
2  a-w*<J  —  10  «/»c'*  +  17  ac  -f-    7  be*  remainder. 

Explanation.  —  Conceiving  4  a^6V  to  have  Its  sign  chanjjed 
to  — ,  and  adding  it  to  (\u'-b^c  we  get  2  a'^b\;  conceiving  8  «/x=^  to 
become  —  3  ahc'^  and  adiilng  tliis  to  —  7  <x6c\  we  have  ^  10  nbc^  ;  con- 
ceiving —  8  ac  to  become  +  8  ac  and  adding  to  4-  9  ac  we  have  +  17  ar; 
finally  conceiving  —  10  be*  to  become  +  10  6c*  and  adding  it  to  —  3  be*, 
we  have  +  7  6c«.     The  difference  is  2  a'^b^  —  10  abc^  -I-  17  ac  +  7  6c*. 

(2.    From   r-.r-  -{-  'A  r.r-  —  r>  c.r  —  4  .r-  take  r^.T^  —  2  r.r 
_^_  .S  ,^x  -  6  ('\ 

(-jr-  -f-  .)  /'./-  —  .»  r.r  —    1  ./  - 

3ear2-Trac  -  4^^  -  ^^a^jqp^ j7« 

Explanation.  — Conceiving  c'-^x^  to  become  —  cV^  and  adding  we 
have  0,  or  simply  c'^z'^  from  cAc^  leaves  nothing.  Next  we  bring  down 
3  ex- since  there  is  nothing  to  be  subtracted  from  it  ;  and  so  in  th«' 
fourth  colunm.  Conceiving  the  sign  of  '^  c'^x  to  become  — ,  and  add- 
ing —  :i  c-x  to  0,  we  have  —  4  c'^x.  Lastly,  changing  —  0  c^  to  +  (\  r-i 
and  adding  to  0  (which  amounts  to  bringing  it  down  with  its  sit/n 
changed),  we  get  4-  <')r'', 

92.   Rule  for  Algebraic  Subtraction. 

1.  Write  the  suhtraliend  under  the  minnend  s(»  that  simi- 
lar terms  shall  stand  in  the  same  eolumn.  It  none  of  the 
terms  are  similar,  unite  the  expressions  after  elianging  the 
signs  of  all  the  terms  of  the  subtrahend. 

2.  Conceive,  in  turn,  the  sign  of  e;wh  term  of  the  subtra- 
hend to  he  changed,  and  ad<l  the  result  to  the  term  above  in 
the  minuend,  writing  the  sum  beneath.  Dissimilar  terms 
in  the  minuend  are  brought  down  with  their  signs  un- 
changed. 


ib  TEXT-BOOK    OF    ALGKBllA. 

93.   Exercise  in  Algebraic  Subtraction. 

1.  rind  the  difference  between  the  following  sets  of 
monomials : 

(1.  8  a  and  6  a.  (5.  16  trb^  and  —  15  a^O'. 

(2.  3  h  and  —2b..  (G.  a-b  and  ab-. 

(3.  -6c  and  11  c  (7.  —  11  acP  and  4  acP. 

(4.  _  14  ^2  j^nd  _  -L3  ^2^  ^g^  43  ^2^4  and  19  a-6*. 

(9.    5  m  and  —  ??i. 

2.  From  2  .r^  -  3  a^x^  +  9  take  cr^  +  5  a^^;^  _  3^ 

3.  From  3  ax  —  A  bfj  -\-  3  cz  take  ax  -\-  3  b}/  —  5  c.^. 

4.  From  V^ab  -2c-l  d  take  2  a;^  -  3  .t^  +  .r  +  1. 

5.  From  «-  +  2  r^r  +  x^  take  a'^  —  x-. 

6.  From  x^  —  3  x-//  +  3  x^t/^  take  —  a'-y  +  T)  j'-//-  —  //. 

7.  Simplify  the  following  expressions,  and  then  take  the 
second  from  the  first :  4  x-f/^  —  5  cz  +  8  ???.  —  4  c.t;  —  2  ???,  and 
cz  -\-2  x^if  —  4  c,^. 

8.  From  a"  —  6  <rr'  +  9  ar}  take  the  sum  of  —  2  a^c 
_  4  «•''  +  2  «r2,  and  -  2  a^c  +  3  «•'  —  r/^^. 

9.  From  a^b'^  +  12  ahc  —  9  ax-  take  4  a^-  —  6  aex  +  3  ^/^^c. 

■     10.    From    x^  ^  ^x^  —  2  x?  -^  1  x  —  1     take    x'^  -\- 2  x^ 

—  2x'-{-  6  .T  —  1. 

11.  A  man  bought  two  storerooms :  for  one  he  paid 
3  ax^  +  2  ar-  —  3  as-  —  ■iat'^  dollars ;   and  for  the  other  ax- 

—  9  as^  +  3  at  —  a^  dollars.     How  much  more  did  he  pay  for 
the  first  than  for  the  second  ? 

12.  A  man  who  had  four  sons  gave  to  the  youngest  2  x 
dollars,  to  the  next  older  2  a  -\- 100  more  than  to  the  young- 
est :  to  the  third  he  gave  2  b  -{-  50  dollars  more  than  he 
gave  the  second ;  and  to  the  eldest  as  much  as  to  the  three 
younger  sons  together.  How  much  more  did  the  eldest  get 
than  the  next  to  the  youngest  ? 


ALGKlilCAlC    Nol  AlloN. 


CHAPTER   VII. 

SYMBOLS   OF   AGGREGATION,   WITH   EXEUCISES   IN 
ADDITION   AND   SUBTRACTION. 

94.  The  Symbols  of  Aggregation  (54,  65)  are  used  to 
show  that  an  enclosed  expression  is  tu  be  regarded  as  one 
number. 

Thus  Qi  a-  -^  li  (lO  -\-  Ir)  is  no  longer  looked  upon  as  three 
terms  to  be  added  together,  but  as  a  single  compound  term. 

We  might  illustrate  this  use  still  further  by  an  example, 
letting  $'M^  stand  for  the  value  of  a  man's  real  estate, 
$2ab  for  his  notes  and  cash,  and  $  b^  for  his  personal 
•property.  Then  f  (3  a^  -\-  2  ah  -f  fr)  indicates  how  much  he 
is  worth  without  any  reference  to  the  values  of  the  differ- 
ent terms. 

Since  the  order  in  which  terms  are  added  is  indifferent 
(28),  a  polynomial  may  be  separated  into  any  sets  of  differ- 
ent terms  by  means  of  i)arentheses,  provided  each  term  is 
included.     This  is  called  the  associative  law  in  addition. 

95.  The  Word  Quantity  as  used  in  algebra  means  an  alge- 
braic expression.  Consecpiently  the  word  quantity  may  be 
used  to  describe  simply  a  positive  or  negative  number ;  or, 
on  the  other  hand,  the  most  complicated  expression. 

a.  Since  (/uuntitt/  and  aU/ehraic  erprexHion  are  synonymous  temis, 
see  again  the  definition  of  the  latter  in  48.  Algebraists,  and  mathe- 
maticians generally,  use  the  word  quantity  constantly.  U  is  there- 
fore very  imjwrtant  lliat  the  student  should  have  a  just  notion  of 
its  signification  in  algebra  as  explained  above,  as  well  as  to  know  its 
original  meaning  given  in  1.  n.     The  wolrd  quantity  is  not  so  suggest- 


<5  TEXT-BOOK   OF   ALGEBKA. 

ive  as  either  of  the  words,  number,  or  algebraic  expression,  and  for 
this  reason  its  use  has  hitherto  been  witliheld.  It  is  hoped  tliat  by 
this  time  the  student  has  sutficiently  clear  ideas  concerning  numbers 
and  algebraic  expressions  that  he  will  be  able  to  use  intelligently 
instead  of  them  the  word  quantity. 


sectio:n^  I. 

On  Compound  Tekms. 

96.  Addition  and  Subtraction  of  Similar  Compound  Terms 
with  numerical  coefficients. 

1.    Add  7  {a+h  -  c),  4  (ci -\- b  -  c),  -  2  {a  ^  b  -  c), 
and  —  Q>  (a  -\-b  —  c). 

Explanation.  —  As  in  63,  6,  the  multiplication  sign  between  7 
and  the  quantity  (a  +  6  —  c)  is  dropped.  Adding  the  coefficients  as 
in  simple  terms,  7  +  4  —  2  —  6  =  3;  .*.  the  answer  is  3  (a  +  6  —  c). 

2.  Add  2  («  +  ^»),  3  (a  +  b),  and  {a  +  b). 

3.  Simplify  4  (a:  +  z)  +  5  (.r  +  z)  —  11  (x  -\-  z). 

4.  Simplify    9  {a  +  bf  +  10   {a  +  bf  -   (a  +  bf 
-2(a  +  by.     (57). 

5.  Add  2  (.x-  +  I/)  +  S  (x  —  y)  and  'd  {x  +  y)  —  2  {x  —  tj). 

6.  Add  4  {ti  +  2  by  -  4  («  -  m),  -  20  {a  +  2  by  + 
5  {a  -  m),  12  (a +  2  by  -U(a-  m),  and  -  5  (a  -\-2by-\- 
5  (a  —  77i) 

7.  Keduce  17  abc  (a*  +  a%  +  crb''  +  «^>"^  -\- b*)  -  6  abc  {a' 
+  a»^>  +  a-b^  +  «i»3  +  b*)  to  its  simplest  form. 

97.  The  Formation  of  Compound  Coefficients  for  a  Common 
Literal  Part. 

1.    Add  ax,  bx,  and  ex  with  respect  to  x. 

Explanation,  a  times  x  plus  h  times  x  plus  c  times  cc  equals  the 
sum  of  a,  6,  and  c  times  x;  or  (a  +  6  +  c)  x.      Otherwise,  ax  +  6a: 

-}■  ex  =  (d  +  h  +  c)  X. 


ALfiEBRAIC    NOTATION.  79 

2.  Add  axy  and  3  jri/  with  respect  to  xy. 

3.  Add  ax  -f-  %?  ''•^'  +  (f'J'   ^^  -^  fih   ^^^^^  i/-^  4-  ^^'J  with 
respect  to  x  and  y. 

4.  Add  aXy  2  car,  and  4  dx. 

5.  Add  a7^i  +  hn,  and  i;«.  -|-  «w. 

6.  Add  ay  -\-  ex,  3  ay  -\-  2  rx,  and  4  y  +  (>  a*. 

7.  Simplify  aar  +  ay  +  rt;^  with  respect  to  a. 

Explanation,     x  times  n  plus  y  times  a  phis  2  times  «  equals 
the  sum  of  x,  y,  and  z  times  (*.     Or,  ax  +  «y  4-  (fz  =  f»  (x  +  y  +  z). 

8.  Add  3  7/12!,  7  7/i//,  —  7  777.^,  and  4  m  with  respei^t  to  m. 
Sue.    3  771X  =  3  X .  7w. 

98.   The  Formation  of  Compound  Coefficients  for  Similar  Com- 
pound Terms. 

1.  Add  a  (x  -\-  y)  and  h  (x  -\-  y).     Answer  (a  -}- b)  (x  -\-  y). 

Note.  —  As  in  other  places  the  multiplication  sign  between  par- 
entheses is  omitted. 

2.  From  a  (x  -\-  y  -\-  z)  take  b  (x  -\-  y  -^  x). 

3.  Add  a  (x  -\-  y)  -\-  b  {x  —  y),  and  77i  (x  -^  y)  —  n  (x  —  y). 


SECTION  II. 

Principles   and    Rules   connected   with  the    Symbols   of 
AormKOATioN. 

99.    Explanation  of  the  Use  of  the  Symbols  of  Aggregation. 

1.  By  using  parentheses  the  addition  or  subtraction  of 
polynomials  can  be  indicated. 

Thus  (-h  5  a^cx'^  -f  4  aHKr^  -f  mxhf)  -(-  (40  ahx^  -  2  a-^  bx' 
4-  6  mx^if-)  indicates  the  sum  of  the  ^wo  enclosed  poly- 
nomials. 


80  TEXT-JiOOK    OF    AL(iEBltA. 

So,  also,  (crb'^  +  12  aba  —  i)  ax'^)  —  (4  ab'^  —  6  acx  -\-  3  a^x) 
indicates  the  difference  required  in  ex.  9,  Art.  93. 

By  the  remark  in  87  the  two  (quantities  given  first  above 
can  be  added  i)y  simply  uniting  their  terms.  AVe  have 
(5  (c^cx^  j^  4  a'^bx^  4-  mx'Y)  +.  (^^>  ((^^-f^^  —  2  (ri^x^  +  0  mxhf) 
=  5  a'^cx^  +  ^  (I'^bx^  -f-  mx~ij^  +  40  a'cx'^  —  2  (i^^it""  +  6  mx^y^. 

It  is  plain  from  this  that  the  only  difference  between  the 
two  modes  of  Avriting  the  sum  is  that  the  first  indicates 
that  it  is  made  up  of  two  parts,  while  the  second  represents 
it  as  one  quantity.  Consequently  if  no  sign  or  a  plus  sign 
precedes  a  parenthesis,  or  other  symbol  of  aggregation,  the 
marks  of  parenthesis  may  be  removed  without  altering  the 
value  of  the  quantity. 

Again,  the  rule  for  subtraction  directs  to  proceed  as  in 
addition  after  liai'ing  changed  the  sign  of  the  subtrahend. 
Hence,  after  changing  the  signs  of  the  subtrahend,  it  may 
be  united  to  the  minuend. 

Thus   (a^b-^  +  12  abc  —  9  ax)  —  ( +  -A  ab'^  —  (3  acx  +  3  a\r) 
=  a^b'^  -\-  12  abc  —  9  ax'^  —  4  ab'^  +  (5  acx  —  3  a'\r. 

This  time  when  the  parenthesis  preceded  by  a  minus 
sign  was  taken  away,  all  the  signs  within  were  changed. 
Of  course  the  reasons  given  apply  to  any  other  polynomials 
as  well  as  to  the  trinomials  used  in  illustration. 

Restating  the  foregoing,  it  is  briefly  this  :  to  add  a  quantity 
(indicated  by  a  plus  sign  preceding  the  parenthesis  enclos- 
ing it)  its  terms  are  joined  just  as  they  stand;  but  to  sub- 
tract a  quantity  the  sign  of  each  of  its  terms  must  be 
changed  according  to  the  rule  for  the  subtrahend  in  sub- 
traction. It  follows  from  this  that  any  terms  in  a  sum  can 
be  enclosed  in  a  parenthesis  with  a  +  sign  before  it  just  as 
they  stand,  but  must  have  their  signs  changed  if  enclosed 
in  a  parenthesis  preceded  by  a  —  sign. 

2.  The  parentheses  are  also  sometimes  used  to  enclose 
the  series  sign  of  a  number. 


AI.CiKHKAIC    NO'IArioN.  81 

Thus,  a  —  ( — b)  nieaiis  that  b  tioiii  the  negative  series  is 
to  be  subtracted  from  n. 

To  remove  such  parentheses,  we  follow  the  same  course  as 
in  the  preceding  case. 

a  -\-  {-\-b)  —  a  -{-  b\   a  -\-  {—  b)  =  (I  —  b\ 
a  —  (-^b)  =  a  —  b;  a  —  (— b)  =  a  -\- b. 

100.  Rules  for  the  Symbols  of  Aggregation — The  previous 
article  shows : 

1.  Any  terms  can  be  placed  within  a  symbol  of  aggregation 
having  a  -f  sign  before  it  just  as  they  stand. 

2.  Any  terms  can  be  placed  within  a  symbol  of  aggregar 
tion  hiiving  a  —  sign  before  it,  if  the  sign  of  each  term  be 
changed  from  -)-  to  — ,  or  from  —  to  -)-. 

3.  A  syml)ol  of  aggregation  having  a  -|-  sign  before  it 
c^n  Ije  removed  without  changing  the  (piantity. 

4.  A  symbol  of  aggregation  with  a  — sign  before  it  can 
be  removed  if  the  sign  of  each  term  within  it  be  changed. 

Note.  —  The  terms  inside  a  symbol  of  aggregation  are  generally 
arranged  so  that  a.  positive  term  comes  first,  when,  following  the 
usual  custom,  no  sign  is  written.  The  sign  before  the  parenthesis  be- 
longs to  the  whole  quantity  inside,  and  not  to  its  first  term.  For 
example,  —  {a  —  b)  =  —  (+  a  —  6);  or,  writing  the  same  expression 
with  a  vinculum,  —  a  —  b  =  —  ■¥  a  —  h.  The  expression  —  (7  —  3 
-I-  4  —  12)  =  —  (—  4)  =  +  4.  Here  the  first  term,  7,  is  positive  The 
beginner  should  not  forget  this. 


SECTION    111. 

ExEiU'isE  IX  Kem()Vi.\<;  thi:  Symbols. 

101.   Exercise  in  Removing  the  Symbols  of  Aggregation  from 
Simple  Expressions.  —  The  results  are  to  be  simplitied  (88.  4). 

1.  3  x'  —  'J  If  -h  CI  ,r  —  1). 

2.  ;■;  .r  +  L>  y  -  (.;•  -  //). 


82  TEXT-BOOK   OF   ALGEBRA. 

4.  2h  -\-ih  -2c)  -\h-^2c\. 

5.  —{a  —  h'l  —  lb-  ^]  -  [c  —  a]. 

6.  —  S  7n  -\-  2  71  —  Sm  —  2  n  -{-  9  m. 

7.  ab  —  (m  —  3  ab  -{-  2  ax)  —  7  ab. 


8.  3a  —  b^lc  —  2a-\-3b  —  5b-^c-\-Zc  —  a. 

9.  1  —  (1  -  a)  +  (  1  -  a  +  ^2)  —  (1  —  tt  +  rt^  -  a»). 
10.    («,  _  .X  _  2/)  +  (x  -  2/  _  ^,)  +  (c  +  2  2/). 

102.  Exercise  in  removing  the  Symbols  from  Complex  Ex- 
pressions. —  The  results  are  to  be  simplified. 

1.    3a -[la -^2  +  J4  6t  -  5  -  (-  2  a  +  9  -  a  -  6)  j] 
-10. 

Let  us  tirst  remove  the  vinculum  by  changing  the  signs  of 
a  and  6,  rule  4. 

3  a  -  [7  a  +  2  +  §4  a  -  5  —  (-  2  a  +  9  —  a  +  6)  J ]  — 10  ; 
Next  we  remove  the  parenthesis,  rule  4. 
3a-[7aH-2-h54a-5  +  2a-9  +  a-6j]-10; 
Next,  the  brace,  rule  3. 

3a  — [7a  +  2  +  4a  — 5  +  2a  — 94-ft  — 6]  — 10; 

Finally,  the  bracket,  rule  4. 

3o^_7a  —  2  —  4a  +  5  —  2a  +  9  —  a  +  6  —  10. 

By  simplifying,  88,  4,  we  have 

8  —  11  a.  Ans. 

2.  2a  -  \2b  -  {3c^2b)  -  a\. 

3.  3a +0^- [a +  5a;  — (3tt  — 2a;)]. 

4.  \-\2-{l-x  +^')S. 

Note.  —  This  method  of  removing  the  symbols,  viz.,  the  inner- 
most first,  then  the  next,  and  so  on,  is  the  best  at  first.  The  expert 
algebraist  writes  the  result  directly. 


Ai>(;i:i'.i:Ai('  notation.  83 


6.    5  «  —  [a  -f  i)  ji  —  >a  —  x  —  3  «  —  2  x}]. 


6.    a  -\-2b  —  \(Sa  —  [3  />  -f-  (8  j-  —  L>  -f-  0  //  —  x)  -f-  4  a] 
—  3b\. 

8.  ax  -\-  ^cx  —  (  mx  -\-  ex  —  y)  -\-  [inx  —  {ex  +  y)]- 

9.  vi^\x-l-4.y  +  2x^{ay-x)  ^p-]. 

10.  {u'bc  +  3  r^)  -f  3  </'-//r  —  {m  —  r-)  —  ;  —  (4  a-^c  -f-  r) 

11.  ff)  a  -  3  />  -  2  r)  -  J5  a  —  (10  6  +  5  r)  5  +  |2  6 

-(2/--L'.o  ;. 

12.  J,, -[A  +  (..  +  !>..)-(  //-..)](. 

SECTION    IV. 

Uses  of  thk  Symbols. 

103.  General  Exercise.  The  Uses  of  the  Symbols  of  Aggre- 
gation. 

1.  AVrite  the  polynomial  n  -\-  b  -\-  c  -\-  d  as  a  binomial  so 
that  the  sums  of  a  and  b  and  c  and  d  may  be  regarded  as 
single  quantities. 

2.  Write  a^  -{.  2  ab  ^  b-  —  r^  —  2  cd  —  d^  as  a  residual 
binomial,  regarding  the  first  three  and  last  three  terms  as 
single  quantities. 

3.  Write  a  -^  b  -\-  e  na  a.  binomial  so  as  to  show  the  sum 
of  a  and  b  as  one  quantity. 

4.  Write  fl2  +  2  ab  -\-  b-  —  2ae  —  e^-^2  bc  —  a-\-b—c 
as  a  trinomial  of  which  the  sum  of  the  squares  forms  one 
term,  the  sum  of  the  })roducts  a  second,  and  the  sum  of  the 
factors  a  third. 

5.  Write  12  nx  +  12  ay  -f  4  by  —  12  bz  —  15  ex  -{-  6  ey 
-f-  3  ez  so  that  the  terms  which  contain  x  may  appear  as  one 
quantity,  the  terms  which  contain  y  as  one  quantity,  and 


84  TEXT-BOOK    OF    ALGEBRA. 

the  terms  which  contain  ^  as  one  quantity,  but  subtracted 
from  the  other  two. 

6.  A  man  who  owns  a  mill  woi-th  $a,  has  personal  prop- 
erty worth  $^  and  has  %d  in  the  bank.  But  he  owes  %e  for 
his  engine  and  owes  his  millwright  $/.  However,  the  mill- 
wright is  indebted  to  him  for  $^  worth  of  flour.  How  shall 
we  represent  what  he  is  worth,  indicating  the  worth  of  the 
mill  as  one  part  and  all  the  rest  as  another ;  also,  keeping 
his  personal  property  separate  from  his  accounts,  and  indi- 
cating the  difference  he  owes  the  millwright  as  one  sum  ? 

7.  Remove  the  parentheses  from  the  preceding  result, 
obtaining  the  man's  fortune  indicated  as  assets  and  liabili- 
ties. 


ALGEBRAIC   NOTATION.  85 


CHAPTER    \  II. 

MULTIPLICATION. 

104.  The  definition  of  multiplication  was  given  in  Art.  34. 
Consult  again  that  article. 

We  have, 
4-3  X  -1-4  =  -f-  11';  .S  X  -  9=  -27;  -4xS  =  -.32; 

-12X  -15=  +1S(). 
-f  J-  X  -\-  ij  =  -\-xi/]  X  X  —  If  =  —  xf/ ;  —  X  X  1/  =  —  x}/ ; 

-  X  X  —  ij  =  -\-  xt/. 

a.  In  Art.  36  there  was  given  an  investigation  of  the  rule  of 
signs  in  nmUiplication.  It  will  be  helpful  to  give  similar  proofs  at 
this  point,  using  letters,  wliich  may  stand  for  any  numbers,  instead 
of  figures. 

1.  To  multiply  +  bby  +  a. 

Once  4-  6  is  +  6;  twice  +  />  is  +  2  6;  three  times  +  h  is+  :i  /.: 
and  so  on.     Then  a  times  +  Ms  +  ah. 

2.  To  nuUtiply  —  d  by  c. 

Once  —  ri  is  —  (/;  twice  —  d  is  —  '2  d;  three  times  —  d  is  —  :i  d. 
Then  c  times  —  dis  —  cd. 

3.  To  nuiltlply  ;j  by  —  in. 

From  what  was  learne  1  in  Chapter  II.,  this  means  that  u  is  i,o 
be  taken  0—  m  times;  i.e.,  uo  times  less  m  times.  Hut,  0  x  «  =  0, 
and  ut  X  n  =  wn.  Subtracting  the  latter  protiuct  from  tlie  former, 
we  have, 

0  —  inn  =  —  inn. 

4.  To  multii)ly  —  n  by  —  m. 

Hy  tlie  same  reasoning  as  before,  multiplying  —  n  by  0  —  />/, 
we  would  have, 

0  —  (—  nin)  =  -\-  nin. 


86  TEXT-BOOK    OF    ALGKBliA. 

105.  Rule  for  Signs   in  Multiplication Like  signs  give 

plus,  and  unlike  minus.  This  rule  applies  to  all  other 
expressions  as  well  as  to  single  numbers. 

106.  Multiplication  of  Monomials In  the  multiplication 

of  monomials,  as  in  the  product  —  5  d^x  X  +  8  cuj,  there 
are  three  things  to  be  considered ;  viz.,  the  signs,  the  numer- 
ical coefficients,  and  the  literal  part. 

1.  The  Signs.  —  The  fundamental  rule  of  signs  is  that 
just  given  in  105.  Extended,  it  includes  the  rules  of  39 
and  40. 

2.  The  Numerical  Coefficient.  —  The  numerical  coeffi- 
cients can  be  multiplied  as  in  arithmetic,  giving  the  numer- 
ical coefficient  of  the  product. 

Thus  T) aX&b  =  r^XC^Xah=:  'M)  ah. 

3.  The  Literal  Fart.  —  We  learned  in  56  and  57  that 
a*  is  the  shorthand  way  of  writing  a  X  aX  aXa\  a^  of 
a  X  a  X  a-,  a^  oi  a  X  a;  or,  dropping  the  multiplication 
sign,  a^  =  aa,  a^  =  aaa,  a'*  =  aaaa,  etc. 

(1.    Let  us  seek  the  product  of,  say,  ah,  a%,  and  ex". 
Writing  out  the  factors,  we  get, 

aXaXcXaX  a  XaXbXcXxXx. 

Or,. by  38,  '2,  changing  the  order  so  as  to  bring  the  same 
letters  together,  we  have, 

aX  a  X  aX  a  X  a  XbX  cX  c  Xx  Xx  =  a^b(fx-. 

For,  a  X  a  X  a  X  a  X  (^  =  (^^^  ("  X  c  =  c^,  and  x  Xx  =  x^. 

And  just  so  it  will  be  in  every  example  of  such  multi})li- 
cation.  A  letter  that  appears  in  two  or  more  factors  is 
repeated  as  many  times  as  indicated  by  the  sum  of  its 
exponents.  A  letter  that  appears  in  only  one  term  is 
repeated  in  the  product  merely  as  many  times  as  there  are 
units  in  its  exponent. 


algebhak;  notation.  87 

(2.    In  the  same  manner  let  us  multiply  a;*"  by  jc". 

x^  =^  X- X  X   X   X  .  .  .  X,  m  factors  being  multiplied 
together. 

x"  =  X- X  X- X  .  .  .  X,   11   factors   being    multiplied 
together. 

x'^x^=xxxxxx  .  .  .  X,  m  -\-n  factors  being 
multiplied  together. 

.-.  x"*  x"  =  x'"  +  "  ; 

i.e.,  the  exponents  of  the  factors  are  added  for  the  exponent 
of  the  same  letter  in  the  product. 

(3.    Multiply  together  the  following  compound  expres- 
sions. 

(a  -J-  by  mx  and  («  +  ^)  (j*^  +  '0  V' 
Writing  the  product  in  full,  we  have 
(«-}-/>)  (rt  -\-h)  {a-\-  h)  {in  -\-  n)  mxy  =  (a  -\-  by  (m  -{■  n)  mxy. 

4.    Examples  of  the  Multiplication  of  Monomials. 

(1.    4  a-  X  0  abr  =  24  a'^bc. 

Reasons.  +  by  +  gives  -f  ;  4  X  6  =  24 ;  a'^  X  a  =  a«, 
or  abiding  the  exj)onents  2  -f- 1  =  3 ;  b  and  c  appearing  in 
only  one  factor  are  brought  down. 

(2.    ~Sxj/Xxt/= -Sxhf. 

Reasons.  —  by  -f  gives  —  ;  3X1=3;  x  X  x  =  x' ; 
.'/  X  /  =  i/. 

(3.    7  m-n'p^  X  -  mn"  X  -  2  nY  X  -  3  jJ"^^  =  -  42 

Reasons. — An  odd  number  of  —  signs,  viz.,  three,  gives 
—  in  the  product ;  7  X  1  X  2  X  3  =  42 ;  m^  X  m  =  m«,  or, 
2  +  1=3;  n'  X  n"  X  ii^  =  7i'+«+^  for  s  +  «  +  2  is  the  sum 
of  the  exponents  of  n:  p'  X  p^  X  p*~^  =  /)'"•"%  for  the  sum  of 
the  exponents  is  ^  4-  2  -1-  r  —  2  =  y?  +  ?'. 


88  TEXT-BOOK   OF   ALGEBRA. 

107.  Rule  for  the  Multiplication  of  Monomials. 

1.  An  odd  number  of  minus  signs  gives  minus  in  the  pro- 
duct ;  otherwise  the  product  is  positive  (39). 

2.  Multiply  together  the  coefficients  which  are  expressed 
in  figures. 

3.  In  the  literal  part,  add  the  exponents  in  the  factors 
for  the  exponent  of  the  corresponding  letter  in  the  product, 
and  bring  down  single  factors. 

a.  For  convenience  it  is  customary  to  arrange  the  literal  part  in 
the  order  in  which  the  letters  come  in  the  alphahet.  Thus,  ax  would 
be  written,  and  not  xa,  unless  for  some  special  reason;  x'kim'^y  would 
be  arranged  into  am^x^y;  zhjx:^  into  x~yz-;  and  so  on. 

108.  Exercise  in  the  Multiplication  of  Monomials. 

1-  (+  8)  (-  2).  8.  _  2  a^  X  -  '^  a%. 

2.  (+6«)(-2a).  9.  a%<^XaH>'. 

3.  (+5wm)  (+9??^).  10.  2xi/X2x'^!/. 

4.  (+  8  ah)  (-  3  c).  11.  -  \  ab-.v  X  T)  ,tx'ii. 

5.  (2  ah'^x)  (3  aVhr'').  12.  -  .vhj'-  X  -  j'Yz-. 

6.  —  xhj  X2  ax.  13.  2  ./•'"  X  x. 

7.  7  a^c  X2ah.  14.  //"  X  —  )/. 

15.  -  3  (a  +  by  X  -  9  {a  +  bf. 

16.  2  (:r  +  y)  X  4  a''  {x  +  y). 

17.  2mXnX  —  a  X  —2  b. 

18.  8  <ib"  X  2  (i-c  X  —  5  c"^. 

19.  -  6  <drt/^  X  2  bif  X  4  (r,i. 

20.  —  3  (ibxii  X  —  2  (I'b.v-  X  —  8  0--  X  —  5  if  X  —  a. 

21.  —  5  (f-b  X  (tb^  X  —  3  «.V  X  —  5  ab(\ 

22.  -  3  (Mm  X  —  2  cV///a  X  -  ^'^  r^tba. 

23.  2  j--^y  X  3  a-?/2  X  X-  X  -  x-  X  -  x^z. 

24.  -  7  anv'  X  -  3  Z^^^i-  X  -  4  «2/,  x  -  a%n  X  -  2  b^n  X 


AL(;K!ii:Al("     NOTATION.  89 

25.  —  e^x  X  —  'SxX  —  Irr  X  —  m/  X  —  r>  rhj  X  —  exy 

Xif. 

26.  \  <ij'  X  '^  rx  X  —  \  mx  X  —  4  if  X  6  m. 
Tl.    x"  X  x^  X  ./"•  X  X''. 

109.    Multiplication  of  Polynomials. 

1.  To  multiply  a  j)olynoiiiial  1)V  a  monomial,  as, 

a  —  //  —  r  -\-  <l  by  ///. 

To  multiply  a  —  h  —  r  -\-  d  by  m  is,  ot  course,  the  same 
as  to  add  m  quantities  eacdi  e([ual  to  a  —  h  —  r  -\-  d.  In 
this  sum  a  would  appear  m  times;  — />,  m  times;  —  <\  vi 
times  ;  and  -|-  </,  m  times.  Hence,  m  (tt  —  h  —  c  -\-  d) 
=  ma  —  mh  —  ?w/*  -|-  md. 

2.  Since  by  the  commutative  law  the  nrdfr  of  multipli- 
cation is  indifferent, 

(a  —  b  —  c  -\-  d)  m  =^  am  —  hm  —  cm  -j-  dm. 

And  so  it  is  in  general.  Whatever  be  the  number  of 
terms  within  the  parenthesis,  each  term  of  the  quantity 
inside  must  be  multiplied  bv  the  quantity  outside. 

3.  To  multiply  two  polynomial  sums,  as, 

a  -\- h  -\-  c  -\-  d  hy  m  -\-  n  -\-  p  -\-  q. 

Here  it  is  plain  that  a  -\- b  -{-  c  -\-  d  is  to  be  taken  the 
sum  of  7/1,  71,  p,  and  q  times.     This  gives, 

m  (a  -\- h  -\-  r  -\-  d)  +  n  (a  -\- h  4- r  -^  d)  -{- p  (a  +  b 
-\-  f  -{-  d)  -\-  q  (a  -\-  b  -\-  c  -\-  d)  =  ma  -\-  vih 
-\-  mr  -f-  wd  -f-  na  -\- nb  -\-  ne  -{•  lul  -(-  p^  +  pb 
_|_  pr  4-  pd  -f  qn  +  qh  -\-qc  +  qd. 

Thus  every  term  of  the  multiplicand  is  niulti})lied  by  each 
term  of  the  multiplier,  Jis  in  multiplication  in  Arithmetic. 
Indeed,  the  explanation  just  given  is  the  justification  of  the 
arithmetical  rule. 


90  TEXT-UOOIv  Oi^  ALGEBRA. 

4.  Finally,  let  us  multiply  two  differences,  say, 

X  —  y  by  m  —  n. 

Now,  X  —  y  taken  m  —  n  times  means  x  —  y  taken  m 
times  less  x  —  y  taken  n  times,  or, 

m  (x  —  y)  —  n  (x  —  y)  =  mx  —  my  —  {nx  —  ny). 

Therefore,  removing  the  parenthesis  in  the  last, 

(in  —  n)  (x  —  y)  =  'i^^^  —  "fi^y  —  nsc  -j-  ny. 

By  inspection  one  sees  that  to  obtain  the  product  both 
terms  of  the  multi|)licand  liave  been  multiplied  by  each 
term  of  the  multi])liei-.  according  to  the  rule  of  signs  (105). 

Thus,  —  my  comes  from  multi])lying  —  y  by  in,  and 
+  ny  from  multi})lying  —  y  hy  —  n.  This  result  may  be 
(H)nsidered  as  a  generalized  proof  of  the  rule  of  signs  (36). 

By  uniting  sets  of  positive  terms  into  one,  and  negative 
terms  into  one  with  parentheses,  any  example  of  the  mul- 
ti})lication  of  i)olynomials  can  be  reduced  to  the  multiplica- 
tion of  two  residual  l)inomials.     Thus, 

((I  —  h  -}-  r  —  (I  —  e  )  (l  —  0  —  ])  —  q  —  r) 

=  [("  +  n  -  (f'  +  '^  +  '^)]  U-("+p  +  q  +  r)] 

=  (//  +(•;/-  {o  +  r)  (o  +  J,  -f  y  +  r)  -  (^  +  f? 
-^e)/  +  (/>  +  d  -f  .)  (o  +  ^.  +  y  +  r). 

Now,  remembering  3,  above,  it  follows  tliat  in  the  final 
product  every  term  of  the  multi])licand  woidd  be  niulti])lied 
by  each  term  of  the  multi})lier. 

5.  The  Distributive  Law.  —  The  multiplication  of  every 
term  of  tlie  multiplicand  by  each  term  of  the  multiplier  is 
termed  tlie  <listriJnifin'  hiir. 

110.    Examples  of  the  Multiplication  of  Polynomials.  —  The 

multiplication  of  |)olynomials  is  indicated  by  enclosing  them 
in  parentheses. 


ALGEBRAIC   NOTATION.  9l 

a.  The  operation  of  inultiplicatioii  is  siinihir  to  that  in  arithme- 
tic, but  with  this  nioiiitication,  that  we  usually  let  the  work  extend 
to  the  right  instead  of  to  the  left.  To  this  end  the  first  term  of  the 
multiplier  is  set  under  the  first  tenn  of  the  multiplicand,  the  second 
term  under  the  second  term,  and  so  on.  Moreover,  we  begin  at  the 
left  to  multiply.  In  the  following  examjjles  the  multii)lication  is  first 
indicated,  and  then  the  work  is  performed  beneath.  As  in  arith- 
metic, the  simpler  of  tlie  two  factors  is  generally  taken  as  the 
multiplier. 

1.    4  a  (3  a  -2  b)  =  \>  2.    (3  i/'  +  //  -  2)  j-i/  =  ? 

3a  -':b  Sf-\-i/-2 

4a xji 

12a^  -  8  ab  3xf  -{- xf  -  2 xi/ 

3.-7  a^cx^  (4ai/  -3x  -\-2  r)  =  ? 
4  ay  —  .*)  ./•  -|-  -  r 
—  7  a*ex'^ 


-  28  a*  cxhj  -h  21  a^cx»  -Ua^  ch 

[.    (3< 

/^  _  4  //^  _  2  ^.2)  X  _  „-2^.  ^  •> 

3  a*- 4  A*  -  2  t- 

—  «'c 

. 

-  3  a*«  -h  4  aVt^e  -f  2  « V 

2 ./ - 

-3.1-4 

2x-' 

-  3  .i-  —  4 

4x*  —  (yx*  —  H  x^  m\ilti[)l yinj,'  by  2 

—  (•)  ./•='  4-  1»  y-  -\-  12  ./•  multiply iiig  by  —  .'! 

—  H  jr'^  -h  12  X  -f  1()  iniiltiplyiii<,^  by  —  4 

4  .r*  —  12  ar«  —  7  r'  -f-  24  r  +  H5  J//.v. 

6.    ( 3/rV;  -  2  <^//  -f-  //*)  (2  (fh  -\-  h')  =  •: 

:\„v,  -  2nh-'-\-h^ 

2ab-\-b-' 


» ./■ 


U/ 


(5  aV>2  —4  aV»*  -\-  2  ab^  multiplying,'  bv  2  <ih 

_  -  2  ab^  -\-  ^f^'b^  -f  ^«       nmltiidying  by  b' 


92  TEXT-BOOK   OF   ALGEBRA. 

Explanation.  —  The  first  term  of  the  second  product  is  3  a^b"^, 
but  it  is  not  similar  to  any  term  in  the  first  product,  and  so  we  set  it 
to  the  right  and  in  a  line  beneath  the  first  product.  The  second 
term  of  the  second  product  is  —  2  ab^,  and  we  set  it  under  the  term 
to  which  it  is  similar;  and  so  generally,  as  soon  as  dissimilar  terms 
appear,  they  are  placed  to  the  right. 

7.    (x^  -3x:'-{-2x  -f  1)  (x^  -  2  ./•  -  2)  =  ? 
^4  _  3  ^2  _^  2  .r  ■+  1 
a.3  _  2  ^  -  2 


x'' 

-3x' 
-  2  x' 

+ 

2  x^  +  x^ 

+  6  x^ 
2x' 

-  4:x'  -2x 

+  6  -y'  -  4  .T  —  2 
x^  —  T) x^  -{-  7  x^  -\-  2 x'^  —  (\x  —  2      Ans. 

8.  (^2  _  „/,  _  f,j.  ^  //2  _  i,^  _|_  ^2)  (^„  -^  />  _}_  c) 

a^  —  ah  —  <ic  -f-  h-  —  Jk'  -\-  r 

a  J^h  -]-  f 

<(^  —  a'h  —  rrc  -\-  aly'  —  ahe  -j-  acr 

-{-  (i"h  —  alt'  —  ahc  +  h^  —  Ire  -\-  be' 

-f-  a'^c  —  nhr  —  ar-  -f-  b'e  —  he"  -f-  e'^ 

^8  -  .S  ahr  ^T/^s  -\-  r^ 

Jn,s. 

9.  (a'  +  ah  +  h"-)  Or  -  ah  -^  fr)  (rr  -  U')  =  ? 

a-  -\-ah-\-  //^ 
a'  ~  ab^h'- 


a* 

+ 

a^b-\- 

+ 

an? 

a"-h"-  - 

a  Jr  -f 

+  />4 

a" 

_ 

+ 

a'¥r 

-/>« 

+  A^ 

a^ 

+ 

aHi'  + 

a^ 



//• 

y^ 

111.   General  Rule  for  Multiplication c 

1.  Monomials.     See  107. 

2.  Polynomial  and  Monomial 


aia;i:iikaic  notation.  93 

Multiply  every  term  of  the  polynomial  l>y  the  monomial, 
according  to  the  rule  for  monomials,  and  unite  the  products. 

3.   Tolynomials. 

(1.  Two  Polynomials.  —  Multiply  every  term  of  the 
multii)licand  by  each  term  of  the  multiplier  and  add  the 
partial  products. 

(li.  Three  or  more  Polynomials.  —  Multiply  any  two 
together  and  their  product  by  a  third,  and  so  on. 

<t.    If  one  factor  of  a  product,  is  0,  of  course  the  product  is  0. 

112.   Exercise  in  Multiplication. 

1.  Multi})ly  together  N  <^  —  0  a.  2  ir.  —  h,  —  M  (fh.  and  0^. 

2.  (2f(  -{-3b)'X2X'2Xr. 

Remark. — Set  down  this  and  the  following  exercises  as  in  110, 
always  combining  monomial  factors  —  when  there  are  more  than  one 
—  into  a  single  monomial  product,  before  using  the  polynomial. 
Thus  the  monomial  factor  here  is  4  c. 

3.  —  'i  nj'  (Jnf  —  2  ex  -j-  />). 

4.  (13  in^n-  —  1 1  nV>'^  -f-  5  x*if^)  X  0  aha% 

5.  Multiply  —  ///  —  n  —  (I  —  c  by  —  m. 

6.  {a-2b-'Sc)  (  -  14  r). 

7.  -{a  -h  -r)  {-  2abr). 

8.  4  ah  (a^  -  :\  ah  -  5  //^)  (-  2  ah% 

9.  Multiply  r  -h  3  by  J-  4-  -• 

10.  Multiply  .r  +  8  by  x  -  2. 

11.  Multij)lyy/-  -|-  7-  by  j-  —  >/'. 

12.  Multiply  X'  -f  f/'^  by  x^  -  y'K 

13.  (x^-  -\x  -\-  16)  (ar  +  5). 

14.  {X  —  9)  {x  —  5). 

15.  X  -{-  a  and  x  -\-  n. 

16.  m  -|-  <'  «iii<l  '"  +  b. 


94  TEXT-BOOK   OF   ALGEBKA. 

17.  5  a?  -f  4  2/  and  3  cc  —  2  ?/. 

18.  a  -{-  b  —  c  and  vi  —  n. 

19.  3  X  -\-  1/  —  3  ,t;  and  a-  +  ^  -f  ^. 

20.  x^  -{-  y  and  it  +  y^- 

21.  x^  —  ic  +  1  and  .r  +  1. 

22.  (2  «c2  -  3  bi/)  (2  c^  -  3  y/'O-      • 

23.  (^^  +  x'y  +  i/)  (x^  —  ir//  +  2/^). 

24.  (r(.*  +  ^/;-V2  +  a')  (a'  +  ^'')- 

25.  c^  —  <^^  4-  c^  —  c  and  c'-^  +  1. 

26.  X  -\-  y  +  w  and  ?/  +  z  +  «r. 

28.  a  -{-b  —  c  and  tt  —  Z>  +  ^• 

29.  a^  —  a^-^a  —  1  by  ^//^  —  ^*  +  1. 

30.  (x*-{-2x^-\-3x'--}-2x-{-l)  (-3x^)  (-2x). 

31.  {«(^  _  (a  -  ^>)  (^>  +  c)}  -b{b-(a-c)}  =  ? 

SuGGKSTioN. — First  perform  the  multiplication  indicated  in 
(a  —  b)  {b  +  c).  The  product  enclosed  in  a  parenthesis  takes  the 
place  of  the  factors.  Then  remove  the  two  inner  parentheses. 
Lastly  b  is  multiplied  into  each  term  of  the  second  bracket  quantity, 
according  to  the  distributive  law,  and  this  product  is  subtracted  from 
the  first  brace  quantity. 

32.  (x  -  1)  (x  -  2)  -  3  .^  (a:  +  3)  +  2  {(x  +  2)  (;r  +  l)  -  3} . 

33.  4  [4  (t  -  (4  ti  +  1)  (a  -3  )  -  9]  -  a  {(12  a  -  2)  2  - 
(a -3)  (a -4.)}. 

34.  a"'-\-b"  by  ff"'-\-b". 

35.  (l  +  ,y  +  y-^  +  y^  +  _,/)(l_y). 

36.  a'-\-a'^-\-a'  by  ^/ +  1. 

37.  (x  -5)(x-  ())  (./•  -  7). 

113.  Use  of  one  Letter  to  stand  for  a  Quantity  expressed  by 
any  number  of  letters. 

In  109,  4,  we  had  x,  y,  m,  and  7i.  each  representing  sums 
of  other  letters.     Likewise,  in  any  example,  one  letter  may 


ALGEBKAIC    NUTATION.  95 

take  the  place  of  any  coiiibination  of  letters.  For,  when  the 
combination  is  reduced  to  its  numerical  value,  it  gives  one 
number,  and  the  letter  used  to  take  the  place  of  the  com- 
bination has  this  number  for  its  value. 

Thus,  A  may  stand  for    ^!—t 

m 

or,  d  may  replace  (3  c^  -\-  7  xyf. 

For,  if  a  =  5,  i  =  4,  c  =  3,  m  =  11, 

a^-h3^>c^_5'^-h3-4-3_61 
m       ~         11  ~11* 

Thus  A  must  have  the  value  — 

In  like  manner,  if  c  =  2,  a-  =  3,  and  y  =  1. 

c/  =  (3  •2»-f  7  •3- 1)2  =  45-^  =  2025. 

<i.    A  rule  or  theorem  expressed  by  means  of  letters  is  called  a 

J'onmdii. 

114.   Theorems  in  Multiplication. 

There  are  certain  problems  in  niultiplicjition  which  occur 
so  many  times  that  a  great  saving  is  made  if  the  learner  be 
able  at  once  to  write  do\\Ti  the  product  instead  of  going 
through  the  operation  of  multiplication.  They  are  called 
theorems  l)ecause  they  are  proofs  of  general  truths  (201). 
The  theorems  of  multiplication  are  of  a  very  simple 
character. 

1.   The  square  of  the  sum  of  two  quantities. 

(1.   Examples.     The  student  will  write  down  and  mul- 
tiply the  following  as  in  112. 

(1)  (a  +  by  =  (a-\-b)(a-\-  b)  (bT),     Am.,  cr  ■\-2ab-\-  b\ 

(2)  {in}^n^)\  (4)    (3a;+4)«. 

(3)  {hd^b^2)\  (6)    (3a«6  +  7c»)«. 

(2.   In  order  to  generalize  this  problem  we  say, 
Let  A  denote  any  quantity  (113), 
and  r>       ''        "     second  quantity. 


96  TEXT-BOOK    OF    ALGEBRA. 

The  square  is  found  by  simple  niulti- 
^J-    '   :?  plication.     The  result,  since  it  is  true  for 

\  i^        A  T^  ^^^  quantities,  may  be  translated  into  a 

T  ^1^    ,   T.2        general  theorem  as  follows  :  — 

A^  +  2  AB  4-  B-^  G^-    Theorem  I.    I'/te  square  of  the 

sum  of  two  quantities  is  equal  to  the 
square  of  the  first,  ])lus  twice  the  product  of  thf  first  hij 
the  second,  plus  the  square  of  the  second. 

(4.    ExEKcisK.  —  The    result    is   to   be   written   down 
without  multiplying. 

(1)  (3  +  4)'-^  =  3--^  + 2   3X4  +  4^  =  9  +  24  +  1(3  =  49  =  7^. 

(2)  (4  ax  +  9  ijf  =  16  a-x-  +  72  </./•//  +  81  ?/. 

(3)  (2  a -\- 4:  by.  (4)    (om-\-2npy. 

2.    The  square  of  the  difference  of  two  quantities. 

(1.    Exami)les.  —  To  be  set  down  and  the  multiplication 
performed  as  in  112. 

(1)  (11 -6f. 

(2)  (a-hy. 

(3)  (a'-2l>)\ 

(2.    Generalization. 


(-t) 

(4  «(2  - 

-  6  »')-. 

(5) 

(fi  x"  - 

-If. 

(«) 

(ix;,- 

-3i 

'■)•-■ 

A  -B 
A-B 


Let  A  denote  any  quantity,  and  B  de- 
note any  second  quantity.  Then  the 
^'  I  AB+  B-^  ^'1''^'"^  of  A  -  B  is  A-^  -  2  AB  +  Bl 
A^_2AB+B^     ^'''''' 

(3.  Thkokem  II.  The  square  of  the  difference  of  two 
quantities  is  equal  to  the  square  of  the  first  minus  twice  the, 
product  of  the  first  by  the  second,  plus  the  square  of  the 
second. 

(4.  Exercise.  —  The  answers  must  be  written  down  by 
the  theorem. 


ALGEBKAIC     NoiATioN.  97 

(1)  {a-'6aby.  (4)   ((>y-7«)«. 

(2)  C^x-Syy.  ip)    (3/-^')'. 

(3)  (4y'-3y.  (6)   (10-D//m)*. 

3.    The    product    of    the    sum    mid    difference    of    two 
quantities. 

(1.   ExAMi'LEs.  —  Set  down  and  multiply  as  in  112. 

(1)  (pa-\-b)ipa-b). 

(2)  (3//r-i-r)/>)(3//i-'-5yv). 

(3)  (11  m'  -  6  w*)  (11  m-  +  6  n'). 

(4)  (15«*-l)(loa^-f-l). 

(2.    Generalization. 

A  +H 

A  —  B  Let  A  denote  any  <piantity,  and  B  denote 

A*  4"  AB  any  second  quantity.     Then  the  i)roduct  of 

-  AB  ~  B«    A -f-B  and  A  -  Bis  A'- B%     Hence, 
A'  -  B' 


(3.  Theorem  III.  The  prod  art  (tf  the  simi  und  differ- 
ence of  two  quantities  is  e<i\nil  in  the  differeiin'  nf  their 
squares. 

■  (4.   ExEKcisE.  —  The  answers  must  be  written  down  by 
the  theorem. 

(1)  (8-h5)(8-r>). 

(2)  (6«6-f  9)(Gr/^-9). 

(3)  (lxyz-\){lxi,z  +  l). 

(4)  (a"  -I-  b")  {a"  —  b"). 

115.    Exercise  in  the  Use  of  the  Theorems  of  Multiplication. 

1.  (a +6)  («+*).  6.    {x-\-y)\ 

2.  (a-f 4)(a-f4).  7.    (3a  +  26)». 

3.  (7  +  5)(7  4-r>).  8.  {n<*d^Ae(Py. 

4.    (a-.r)(n-x).  9.    (\  m^ -^  \  n^. 

6.    (m  -\-7i)  (m  —  v).  10.    (4r/7;*  -  \)\ 


98  TEXT-BOOK   OF    ALGEBKA. 

11.  (6 +  3)  (6 -3).  14.    (imhi-iy. 

12.  (4  a-' 4- 3^-)  (4  a- -3  6').  15.    (a\-f,')\ 

13.  (8  abc  —  9  ab-cy.  16.    (a-"'  —  by)\ 

17.    {a"'-^b"y-. 

116.    Other  Noteworthy  Cases  in  Multiplication. 

1.  Prove  by  Theorem  I.  that 

(a-\-b  +  cy  =  a'^b'^  c-  -\-2ab-\-2ac-\-2  be. 

We  have,    \_{a  +  ^')  +  cj  =  {a^by^2  {a  -\-b)c-\-  &=  a" 

-^2ab-\-b'''^2ac^2bc^c\ 

2.  Prove  by  the  same  theorem  that 

{a^b  ■^c-\-d)-  =  (r  +  b'^  +  c"  +  ^-  +  2  ab  -\- 2  ac 
-{-2adi-2bc-i-2bd-\-2  cd. 

lia^b)  +  {c  +  d)J={a^b-f^2{a^b){e^d) 
J^(c^df  =  a''-]-2  ab  +  b'  +  2  ac^2ad  -^2  be 
-\.2bd-\-c-  +2cd  +  d\ 

The  formulse  of  this  and  the  preceding  example  general- 
ized give  the  following  theorem. 

3.  Theorem  IV.  The  square  of  the  sum  of  any  number 
of  quantities  is  equal  to  the  sum  of  their  squares,  plus  twice 
the  product  of  each  quant  it  tj  by  all  those  that  follow  it. 

4.  Prove  by  aetnal  multiplication  that 

(a  +  by  =  {a  +  b)  (a+  b)  (a  +  b)  =  a^  +  3  a%  +  3  ah^  +  b\ 

From  this  formula  we  have, 

5.  Theorem  V.  The  cnhe  of  the  sum,  of  two  quantities  is 
equal  to  the  cube  of  the  first,  pdus  three  times  the  square  of 
the  first  times  the  second,  plus  three  times  the  first  times 
the  square  of  the  second,  pdus  the  cube  of  the  second. 

6.  {a  4-  by  =  a^  +  4  aH)  +  Ga'"'  li'  +  4  a//  -\-  b\ 

Prove  this  equality  by  actual  multiplication  of  the  former 
result  {(I  -j-  by  by  tt  +  ^.     State  a  theorem  for  this  case. 


ALGKBliAlC    NOTATION.  99 

7.  Perform  the  multiplications  indicated  in  the  following: 

(1.  (x  +  3)(x-h4).  (7.  (x-h2)(a:-6). 

(2.  (x-f6)(u:-hy>  (8.  (x-3)(x-4). 

(3.  (X  -f  13)  (x  -h  17).  (9.  (X  +  9)  (x  -  12). 

(4.  (X -hi)  (or -5).  (10.  (x  -  15)  (a- +  16). 

(5.  (X  +  18)  (x  -  4).  (11.  (x  —  12)  (x  -  8). 

(6.  (x  -h  20)  (X  -  4).  (12.  (X  -  25)  (x  -  30). 

8.  TiiKMKKM  \'I.  77n'  prodKct  of  two  binomials  Jnirlnfj  a 
term  common  is  equal  to  the  square  of  the  common  term, plus 
the  SUM  of  the  other  terms  times  the  common  term,  plus  the 
PRODUCT  of  the  other  teiins. 

9.  Exercise  in  the  use  of  the  theorems  of  this  article. 

(1.    (x  -h  11)  (x  -h  13).  (6.    {a-b-\-  S)\ 

(2.    (x-6)(x-l).  (7.    (Aax-\-5b  +  6cy. 

(3.   (x-20)(x  +  8).  (8.    (15x-9y- 12«)'. 

(4.    (x -h  100)  (x  -  2).  (9.    {Sm-\-n-{-2p-\-  1)'. 

(5.    («  +  //  +  !)'.  (10.    (a  +  2^»-(3c-</))^ 

(11.   (x-h3)(x-h4)(x-3)(x-4). 

Suggestion.  —  Multiply  tl»«'  first  ami  third  factors  together,  and 
the  second  and  fourth,  and  then  tliese  products. 

(12.    (1 +r)(l-f- <•)(!-'')  (l+<'-0- 
(VX    (x-f2//)». 

117.  Simple  Powers.  —  See  detinition  in  57  and  treatment 
(►f  ])owers  of  sim})le  algebraic  numbers  in  42. 

1.   Powers  of  Monomials. 

(1.   Find  the  fourth  power  of  3  a*b^c. 
By  the  rules  of  multiplication  for  monomials, 

3  a*b*c  X  3  a*b^c  X  3  a'b'c  X  3  a'  O'r  =  81  a»*V. 


100  TEXT-BOOK  OF  ALGEBRA. 

(2.    Rule. 

(1)    SiGxs.  — The  rules  of  42,  1  apply  equally  here. 

(2)  The  coefficient  expressed  in  figures  is  raised  to 
tiie  power  indicated. 

(3)  By  the  rule  for  multiplication  (107)  the  expo- 
nents of  the  factors  are  added ;  but  here  the  numbers  to  be 
added  are  equal,  and  we  can  inult'iply  each  exponent  by  the 
number  denoting  the  power  to  which  the  quantity  is  raised. 

(o.      EXEKCISE. 

(1)  Multiply  3  (2  ciH^y  by  4  (  -  ali'ry. 

(2  a^hy  =  4  any';  and  (  -  ab^^  =  -  rrV/v-«. 

.-.  we  have  3x4  a'b'  X  —  4  am-'  =  —  48  «V/rl 

Notice  tiiat  the  coefficients  outside  of  the  parentheses  are 
not  affected  by  the  exponents. 

(2)  Square  the  following  : 

3  xy'\  bx*,  —  3  ex^,  —  2  abc^,  4  a*b^x'^,  a'",  x'-p. 

(3)  Cube 

2  ab-'n%  -  3  a^b,  —  2  ab'iinrK  7  /yy-zr^  a>"b\ 

(4)  Develop  the  foUowiug: 

3  (4  xy,  (-2a.V)^  (  -  ^rxf,  (  -y../)«,  2  (  -  by,  \(ay}\ 

(5)  Develop  and  multiply, 

7  (  -  2  .r//)'  X  (  -  2  am)  ^;    (  -  1)  (  -  l)"^  (  -  1)M  "  Y' 

Kkmahk. — Since  1  =  1-=  1^  =  1+  =  1^,  etc.,  we  may  regard  las 
liaviiig  any  exponent  desired. 

2.    Powers  of  Polynomials. 

(1.  Square  of  Polynomials.  —  See  Theorem  IV.  of 
the  last  article.  Being  true  for  any  number  of  terms,  it 
includes  Theorems  I.  and  11.  of  Art.  114. 


^^  G^''^^^ 


ALGKHHAK;    NoTATIoyV  \^\ 

(2.  Cul)e  and  Fourth  Power  of  Polynomials.  —  See 
116,  5  and  6  for  cube  and  fourth  power  of  a  binomhd.  The 
further  investig^ation  of  the  cases  already  given  and  of  the 
development  of  hi«]:her  powers  is  remanded  to  a  subsequent 
cha]>ter  on  Involution. 


^rr 


fVid 


102  TEXT-BOOK  OF  ALGEBRA. 


CHAPTER   IX. 

DIVISION. 

118.  The  definition  of  division  was  given  in  Art.  43. 
All  the  rules  for  signs  were  explained  there. 

119.  Division  of  Monomials. 

1.  The  signs.  —  See  43. 

2.  The  Coefficients.  —  Divide  the  coefficient  of  the  divi- 
dend by  the  coefficient  of  the  divisor  to  obtain  the  coefficient 
of  the  quotient. 

3.  The  Literal  Part.  —  In  multiplication  we  added  the 
exponents  of  the  same  letter  in  the  factors.  Thus,  a^  X  a^ 
=  a'.  Hence,  by  the  definition  of  division  (43),  we  must 
subtract  the  exponent  in  the  divisor  from  that  in  the  divi- 
dend for  the  exponent  in  the  quotient. 

Thus,  a'  -^a^  =a^',  x"^  -^  x""  =  x"*"". 

Three  special  cases  deserve  attention. 

(1.  JVhen  a  letter  is  found  in  the  dividend  and  not  in 
the  divisor,  it  must  appear  in  the  quotient,  else  when  the 
divisor  and  quotient  are  multiplied  they  will  not  give  the 
dividend. 

(2.  When  a  letter  contains  the  same  exponent  in  both 
dividend  and  divisor  it  gives  rise  to  the  factor  1  in  the 
quotient,  and  is  not  written ;  for,  any  factor  is  contained  in 
itself  once. 

Thus,  Qa^b-^2a^=Zl-b  =  3  b. 


ALGEBRAIC   NOTATION.  103 

(3.  When  a  letter  is  found  Imving  a  greater  exponent 
in  the  divisor  than  in  the  dividend,  or  appearing  in  the 
divisor  and  not  in  the  dividend,  the  quotient  is  usually 
written  as  a  fraction.  A  factor  in  the  dividend  cancels  an 
equal  factor  in  the  divisor,  as  in  the  previous  case.  Hence, 
we  subtract  the  exponent  in  the  dividend  from  that  in 
the  divisor,  and  write  the  letter  with  this  exponent  in  the 
denominator.  When  a  letter  is  found  in  the  divisor  and 
not  in  the  dividend,  it  is  written  in  the  denornirmtor  of  the 
quotient. 

A  fraction  as  here  used  is  to  be  regarded  as  an  expressed 
division,  the  numerator  being  the  dividend  and  the  denom- 
inator the  divisor. 

4.    Examples  in  the  Division  of  Monomials. 
(1.    (\  a*b' -^  2  a'b  =  :UrlK 

Reasons.     -|-  -t-  +   giv<'«   -j-,    (j  ^ '2  =  3',     4-2  =  2; 
2-1  =  1. 
(2.    9  m,'' Ji^j,  ^  3  n'^  =  3  vi'^  p. 

^  4  s 

(4.    -  45  <i*tA'  -^  15  a'b-  =  -  —  . 

a 

120.    Rule  for  the  Division  of  Monomials. 

1.  The  Signs.  —  Like  sign3  in  the  dividend  and  divisor 
give  plus  in  the  quotient,  unlike  signs  give  minus. 

2.  Numerical  Coefficients.  —  Divide  the  coefficient  of  the 
dividend  by  that  of  the  divisor  for  the  coefficient  of  the 
quotient. 

3.  The  Literal  Part.  —  Subtract  the  exponent  of  any 
quantity  in  the  divisor  from  its  exponent  in  the  dividend 
for  its  exjKment  in  the  quotient.  When  the  remainder  is 
negative  the   factor   is  written  in  the  denominator  of  the 


104  TEXT-BOOK   OF    ALGt^^BliA. 

quotient.  Cancel  quantities  having  equal  exponents,  and 
bring  down  the  other  qu.mtities,  if  any,  in  the  dividend 
(numerator)  and  divisor  (denominator)  of  the  quotient. 

121.    Exercise  in  the  Division  of  Monomials. 

2.  -  (U  ^  16. 

3.  48  a%  -h  2. 


5.    12  «V/'  ^  -  3  a-h. 


10  e~x^y 


—  2  evy^ 
-  2  acx^y^v- 


;j  a'c''      ' 

10. 

ah 

11. 

-  21  hhx 
IJhex^'     ' 

12. 

—  (;  (a  —  /,f 

-  2  {a  -  h) 

IS 

6  (m  +  n)^ 

;i  (in  +  ?0' 

14. 

a;-^  -  y' 

X  -y 

15 

(X  -  yy 

-14  ax^y^  (x  —  y)^ 

8.    loaVy  -^  -  10  r/-.r^/A        16.    /''  +-A)''  . 

17.  Ua"'h"cr'  -^ 2  m^^^V. 

18.  —  KLr'y  -^ 8:r*. 

19.  (.r  -  -yf  (m  -  ny  _ 

(x  —  //)"^  o>?  —  7iy 

122.    Leading  Letter  in  a  Polynomial. 

In  most  polynomial  expressions  in  division  there  is  what 
is  called  a  leadhig  letter. 

Thus,  in  ax^  +  hx}  +  ^'•^'^  +  ^^^  +  ^•^"  +/^  +  (h  ^  is  the 
leading  letter.  Usually  it  is  quite  easy  to  distinguish  it. 
Sometimes,  however,  one  letter  seems  as  ])roniinent  as  an- 
other, as  in  a^  -|-  2  ah  +  ^"^  +  2^/^  +  2  he  ~\-  o-.  In  such  cases 
it  is  convenient  to  regard  that  l^tt^r  which  comes  first  in 
the  alphabet  as  the  leading  one.  and  to  let  the  others  come 


ALGEBRAIC    Nn  lATlON.  *  lOo 

after  it  as  nearly  as  may  be  iii  their  alphabetical  order. 
Thus,  a  would  l>e  the  leadinj^  letter  in  the  above  expression. 
In  the  terms  in  whieh  a  does  not  appear  0  would  l)e  the 
leading  letter,  and  so  on. 

123.  A  Polynomial  is  said  to  be  arranged  with  reference 
to  the  powers  oi  the  leading  letter  when  the  exponents  of 
this  letter  either  increase  or  decrease  in  regular  order. 

For  example, 

2  a*  -I-  2  a-'h  -h  r»  a-h'  -  (>  al/'  -f-  4  //* 

is  anunged  with  reference  to  the  descending  powers  of  a. 
Also,  1  -|-2.i-r,. 1-3 +  8.1-5 

is  arranged  with  reference  to  the  ascending  powers  of  x. 

124.  Division  of  Polynomials.  —  Explanations. 

There  are  two  ca.ses  corresponding  To  short  and  long 
division  in  arithmetic. 

1.  Division  of  a  Polynomial  by  a  Monomial. 

As  in  multiplication  we  multiply  every  term  of  the  poly- 
nomial by  the  monomial;  so  in  division,  for  similar  rea.sons, 
we  divide  every  term  of  the  i)olynomial  by  the  monomial. 
Such  examples  may  be  set  down  as  in  short  division  in 
arithmetic. 

3  ,,.'//)  ()  ,,7,2  -f  15  a-b^'c  -  3  a'-be' 

2.  Division  of  a  Polynomial  by  a  Polynomial. 

(1.  Investigation  of  tlie  Process  of  Division.  —  Since 
division  is  the  ccmverse  of  multiidication,  let  us  nndtiply 
two  iK)lynomials ;  then,  using  the  product  as  a  dividend, 
and  one  factor  as  the  divisor,  let  us  try  to  obtain  the  other 
factor  as  quotient  by  retracing  our  steps  in  multiplication. 


106  TEXT-BOOK  OF  ALGEBRA. 


2  a;2  -  3  ax  +  62 
Sx^  +  4bx-  a2 


Qx^-9ax^  +  S  h^x^ 

+  8  &a;3  -  12  ahx"-  +  4  Wx 

-  2  a2a;2  -f  3  gSa;  -  ^252 

6  a;4  -  9  ax^  +  3  ¥x'^  +  ^hx^  —  12  a6a;2  +  4  6%  -  2  a^a:^  +  Z  a^x  -  u^b'^ 

For  reasons  given  farther  on,  we  arrange  the  dividend  with  refer- 
ence to  X. 

Divisor.  Quotient. 

2  a;2  —  3  ax  +  b^)  Dividend.  (3  x^  ■^4bx  —  a^ 

6x*-9  ax^  +  8bx^-2  cfix^  —  12  abx-  +  3  b'^x:^  +  3  a^ic  +  4  b^x  -  ofil>^ 

6  X*  -  9  ax3 +  3  b^x^ 

8  6x3  —  2  a^x^  —  12  a6x2                 +  3  a%  +  4  bH  —  d^U^ 
8  6x3 -  12  a6x2 +  4  68x 

-  2  d^x^  +  3  a^x  -  a-^62 

-  2  a2x2 +3a3x -  d-b^ 

Explanation. — We  know  in  advance  that  the  quotient  is  the 
former  multiplier,  3  x^  -f-  4  6x  —  «-. 

In  order  to  obtain  the  first  term  of  the  quotient,  3  x-,  the  first 
term  of  the  dividend^  6x\  is  divided  by  2x2,  the  first  term  of  the 
divisor.  Then,  as  in  arithmetic,  the  divisor  is  multiplied  by  the  first 
term  of  the  quotient,  giving  6x^  —  9  ax^  +  3  6^x2,  and  the  product  is 
subtracted  from  the  dividend.  In  order  to  obtain  the  second  term  of 
the  quotient,  4  6x,  we  divide  the  first  term  of  the  remainder,  8  6x3, 
by  the  first  term  of  the  divisor,  2  x2.  Then  we  multiply  the  whole 
divisor  by  this  quotient  term,  and  subtract  the  product  from  the  first 
remainder. 

To  get  the  third  term  of  the  quotient  the  first  term  of  the  last 
remainder,  —  2  d^x!^  is  divided  by  the  first  term  of  the  divisor. 
After  multiplying  the  whole  divisor  by  the  quotient  term  just  found, 
and  subtracting,  there  is  no  remainder,  and  the  division  is  completed. 

a.  Remark  on  the  Process  of  Long  Division.  —  If  we  look  closely 
at  the  example  given  above,  we  see  that  the  process  of  division  suc- 
ceeds in  separating  the  dividend  into  three  trinomial  parts,  into  each 
of  which  the  divisor  is  contained  exactly. 

The  same  separation  of  the  dividend  is  seen  in  long  division  in 
arithmetic. 

25)  675  (27        i.e.        25)  500  +  175  =  675 
^^  (Q)  20  +      7  =  27. 

175 
175 


ALGEBRAIC    NOTATION.  107 


(2.   Divide  4x^-x*-\-4xhy2x*  +  '3x-\-2. 

DiVIDKND. 

4  a-6  —  ar»  -I-  4  ar        |  2g'-f  3g  +2     Divisor. 


4x*-|-Ga-*  +  4ar'      2x^—3x*-\-2x     Quotient. 


—  6x*  —  9x^  —  (jx^ 

4ar»H-6ar='  +  4x 

Here  the  dividend  is  separated  into 

(4a:»  +  6ar*  +  4a-«)  +  (-(>x*-9a-«-(>x')  +  (4a:»  +  6x'+4a:). 

By  adding  these  we  may  verify  that  their  sum  is  the 
dividend.  Writing  the  dividend  in  this  form  and  dividing 
as  in  short  division 

2x»  -  Sj--  +  2x 

b.  Necessity  of  dividing  the  first  term  of  the  remainder  by  the 
first  term  of  the  <li visor. 

From  all  that  has  Iwen  said  it  is  plain  that  the  first  term  of  the 
dividend  antl  of  each  remainder  corresjxmd  to  the  firnt  term  of 
the  divisor,  and  when  divided  by  it  give  the  diflferent  tenns  of  the 
quotient.  Now  not  only  the  dividend  but  each  successive  remainder 
must  be  kept  arranged  with  reference  to  the  leading  letter.  Had  we, 
by  a  mistake  in  the  arrangement  of  the  first  remainder  above, 
brought  —  12  nhx'^  in  advance  of  8  bx^,  we  should  have  obtained 
—  0 a/>  as  a  tenn  of  the  quotient! 

125.   Examples  in  the  Division  of  Polynomials. 

1.  Divide  :i  aH*  -f  (>  a*b"  —  3  ab^  by  —  .'i  ab\ 

—  3  ab')  3  a"'b^  -\-  6  a*b''  —  3  ab^ 

—  a'"-^b  —  2a^b»-^-\-b     Ans. 

2.  Divide  4  y*  —  5  y*  + 1  by  1  —  if—  y. 

Ari-anging  lK)th  dividend  and  divisor  afCH)rding  to  the 
ascending  jK)wers  of  y,  we  liave 


108  TEXT-BOOK   OF   ALGEBKA. 

1  —  5  7/2  -I-  4  ?/4    I    1-f-?/  —  2//-       DIVISOK. 

1  +  //  —  2  v/^  1  _  y  _  2  if-     Quotient. 

—  y  —  3  y'^ 

«.  Had  both  dividend  and  divisor  been  arranged  according  to  the 
descending  powers  of  ?/,  it  would  merely  /«rtre  reversed  the  order  of 
the  terms  in  the  quotient.  Sometimes  one  arrangement  is  preferable, 
sometimes  another. 

h.  In  writing  the  partial  products  it  is  often  difficult  to  have  terms 
come  under  those  to  which  they  are  similar.  When  dissimilar  tenns 
are  written  in  the  same  column,  care  is  necessary  on  the  part  of  the 
beginner  to  keep  from  subtracting  their  coefficients. 

c.  The  divisor  and  quotient  of  the  first  example  of  the  preceding 
article  were  written  respectively  at  the  left  and  right  of  the  dividend 
as  in  arithmetic.  For  convenience  of  multiplying  in  Algebra  both 
are  written  at  the  right. 

d.  When  there  is  no  term  in  the  minuend  similar  to  a  term  in  the 
subtrahend,  the  latter  is  subtracted  from  zero,  and  according  to  tbe 
rule  for  subtraction,  is  brought  down  with  its  mjn  changed. 

3.    Divide  d^  +  1  by  d  +  1. 

d'^1       \d±l 


d' 

d^  -  d' 

+  d'  +  d 


d-\-l 


d  +  1 
d-1 


As  in  arithmetic,  the  remainder  is  written  over  tlie 
divisor  and  added  to  the  other  terms  of  the  quotient.  In 
this  addition  a  -|-  sign  must  be  used.     See  52  b. 


ALGKHKAIC    Nnl.vrioN.  lUU 

126.  Rule  for  Division. 

1.  Moiioiiiiiils.     See  120. 

2.  Polynomial  by  a   Monomial.  —  Divide  each  term   of 
the  Polynomial  by  the  Monomial. 

3.  Polynomial  by  a  Polynomial. 

(1.  Arrange  both  diviiU'nd  and  divisor  according  to  the 
descending  or  a-seending  powers  of  the  same  leading  letter. 

('J.  Divide  the  tirst  term  of  the  dividend  by  the  lirst 
term  of  tlie  divisor  and  place  the  resnlt  in  the  quotient ; 
multiply  the  divisor  by  this  quotient  term  and  subtract  the 
product  from  the  dividend,  and  arrange  the  remainder  with 
reference  to  the  leading  letter. 

(li.  Divide  the  first  term  of  the  remainder  by  the  first 
term  of  the  divisor  for  the  sec(md  term  of  the  quotient ; 
multiply  the  divisor  by  this  term  and  subtract  as  before  ; 
and  so  continue  until  all  tlie  terms  are  brought  down. 

(4.  If  there  is  a  remainder  write  it  over  the  divisor, 
and  add  this  fraction  to  tlie  other  terms  of  the  quotient. 

127.  Exercise  in  Division. 

1.  :U)a*h  -. 5«-/i  =  ? 

2.  27  m'n*  (a  -\-  i-y  -4-  9  mu*  (a  -|-  rf. 

3.  3x"*^"  -^  .f'"-". 

4.  a'  -J-  <('. 

6.    (So  mhj  -\-  28  vrtf  —  14  /////')  -. 7  my. 

6.  (12  ay  —  10  a^if^)  -h  4  ct'if': 

7.  Divide  'Myp^q'h'K^  —  7'2  jrf/'-rf^  —  S-ij^f/'-rx-  by  —  12  /»(/-. 

g    4fr6c-f  8  tf&r^-f  12«^V 
2abc 

9     27  («  -  by  -  18  (g  -  by  +  ^^ja  ~  bf 


110  TEXT-BOOK    OF    ALGEBRA. 

10.  Divide  2  x^  -\-  7  xi/  -^  6  y"^  hj  x  -\-  2  f/. 

11.  Divide  ac  -\-  be  —  ad  —  bd  by  a  -\-  h. 

12.  (12  -  4  «  -  3  a^  +  iv")  -V-  (4  -  a^). 

13.  (4  a^  -  5  aH>'  +  ^^4)  -^  (2  a-  —  3  ah  +  Z-/^^. 

14.  Divide  5  .r-y  -f  y'  -j-  x^  +  '">  ^y^  lL)y  A:xy-\-i/-\-  x^. 

15.  Divide  1  a:*^  +  II  ic'^  -  :^  X  +  5  by  -  ic  +  3. 

36  4^4*^3^ 

16.  Find  the  multiplicand  when  the  product  is  3x^ 
-\- lA:  x^  -\- Si  x  -\- 2  and  the  multiplier  is  o;^  -f  5  ic  +  1. 

17  (a-*  —  ^'')  -=-(«-  if). 

18.  (a^  -  243)  -^  (^/.  -  3). 

19.  Divide  a^  —  3  if'*  by  a  -f  a:. 

20.  Divide  G  a V  —  14  «5.r«  +  12  a^x^  —  a''  by  2  aV  _  ^s 

21.  Divide  ax^  —  air-  +  IP-x  —  x^'  by  {x  -\-  b)  (a  —  x). 

22.  (a^  4-  ^»3  +  c^  —  3  a/>c)  -~  (a'^  +  Z*'^  +  c^  —  Z»c  —  ac  —  aZ»). 

/l  1  7  3    > 

23.  K'  ^''  +  2  ^"^  "^  ^'''  ~  '^'  "^  2  '^''  ~  2  '"^ 

"^  V2       2^2 

24.  Divide  x^"  +  x-"i/-"  +  //«  l)y  ic-«  -  x"f/"  +  y-". 

25.  Divide  x"^  —  2  x'  +  1  by  x'^  —  2  a?  +  1. 

26.  Divide  x""  -  a'^  by  x-  -f-  2  ^/o-^  +  2  a.^x  -\-  a\ 

27.  Divide  a'  +  aV/'  +  aV/  +  a-b''  +  ^»«  by  a'  +  a'Z/  +  a-Z*^ 

-^ab''-\-b\ 

128.  Zero  and  Negative  Exponents.  —  Theorems  of  Nota- 
tion. 

Zero  exponents  and  negative  exponents  arise  when  the 
exponent  in  the  divisor  is  equal  to  or  greater  than  that  in 
the  dividend. 

Thus,  a^-ira'  =  a'-^  =  «<>,  and  b^  -j-  b^'^  ^  ?/-i<>  =  b-". 


ALGEBKAIC   NOTATION.  Ill 

Their  use  was  obviated  in  119,  .'i,  as  explained  in  (2  and 
(3  of  tliat  Article. 

1.  Zero  Exponents. 

Let  A  be  any  (juantity  and  n  \w  any  exponei 

Then,      Al  =  A"-"  =  A«> ;  but  —  =  1, 
A"  A" 

.-.  A**  =  1. 

Thus,       50  =  1,  VI''  =  1,  1(X)0<>  =  1. 

Thkorkm  T.  Avij  qiuintity  irhose  exj)o?ie7it  is  0  is  equal 
to  unit  I/. 

Note.  —  There  is  a  sliarp  distinction  between  zero  used  as  a  co<?f- 
ficient  and  zero  used  as  an  exponent. 

For,  Ox  a  =  0,  while  a^  =  1. 

The  zero  exponent  shows  that  the  given  factor  cancels  out  of  tlie 
quotient,  and  is  therefore  used  no  times  as  a  factor  in  it. 

Forexample,    ^^'^  =^l^  x  ^  =  ^J^  x  1  =  ^J^ . 
*  lia^a  lid         a^        lid  3d 

2.  Negative  Exponents. 

In  Article  119  it  was  noted  that  when  the  exponent  of  a 
letter  in  the  divisor  is  greater  than  that  of  the  same  letter 
ill  the  dividend,  the  dividend  cjuantity  is  cancelled  out,  and 
the  letter  with  the  excess  exj)onent  remains  in  the  divisor. 


Thus, 


3  aV,*         h 


But,  now,  if  the  rule  to  subtract  the  exponent  of  the 
divisor  from  that  of  the  dividend  were  followed  strictly,  a 
negative  exponent  would  result. 

Thus,  |^  =  2«»4-«        •  ^ 


112  TEXT-BOOK  OF  ALGEBRA. 

This  suggests  the  idea  that  changing  the  sign  of  the  expo- 
nent removes  the  factor  itself  to  the  opposite  term  of  the 
division. 

To  test  this  a  means  must   be  found  of  transferring  a 

factor  from  one  term  of  the  division  to  the  other.     If,  now, 

5  a^})^ 
we  seek  to  remove  such  a  factor  as  a^  in  — — „—  from  the 

numerator,  it  may  be  done  (using  a  principle  well  known  in 
arithmetic)  by  dividing  botli  terms  of  the  fraction  by  a^. 
(See  156.) 

A'" 
Similarly,  V~"  in — y-~  may  be  transferred  by  m  iltiply- 

ing  both  terms  by  C". 

A'"  A'"  X  C«  A'"C"        A'«C"       A'«C" 


BC-"        BC-^XC«        F>(V'-"        BC^  1 


In  like  manner,  any  other  factor  could  be  transferred 
from  tlie  denominator  to  the  numerator  or  from  the  numer- 
ator to  the  denominator,  by  changing  the  sign  of  its 
exponent. 

Theorem  II.  Any  factor  ukii/  he  transferred  from  one 
term  of  a  division  to  the  other,  providimj  the  sign  of  its 
exponent  is  changed. 

Kemakk.  —  Here  we  s^e  the  opposite  nature  of  the  signs  when 
prefixed  to  exponents.  If  a  poa'itixe  exponent  denotes  that  its  quan- 
tity is  to  be  multiplied  into  another  quantity,  a  negative  shows  that 
its  quantity  is  to  be  used  as  a  divisor  of  the  other  quantity. 

The  student  is  asked  to  note  that  we  have  come  quite  naturally  to 
tlie  meaning  of  a  negative  exponent.  In  subtracting  exponents  we 
have  simply  followed  the  rule  for  division  (which  in  turn  came  from 
the  rule  for  multiplication),  and  have  merely  subtracted  a  greater 
exponent  from  a  less. 


ALGKBKAIC    NoiATloN.  113 


a.    The  following  (U'tiiiition  may  prove  helpful  to  the  student. 

a)  .      integral  power  of  a  number  is  the  continued  ) 

I  negative  j  quotient 

of  1  by  that  number. 

h.   The  Reciprocal  of  a  quantity  is  1  divided  by  that  quantity.    The 

1  1       a~^ 

reciprocal   of   «  is      .     But  by  this  article —•= -- —  =  a~^  .    Hence 
a  a        1 

the  exponent  —  1  denotes  the  reciprocal  of  a  quantity. 

3.   Exercise  in  the  use  of  Zero  and  negative  Exponents. 
(1.    What  will  x"~'^  equal  when  w  =  2  ? 
(2.    AVhat  does  a^  X  «"*  X  a"*  equal  ? 

(3.   Reduce  the  expression  — - — -—  to  an  equivalent 

6  arm~'U 

one  having  positive  exponents. 

By  cancelling  the  a^s  and  transposing  those  letters  which 

have  negative  exponents  U)  the  opposite  term  of  the  division, 

we  derive 

Gahn-^d       (ybe*d 

(4.  3+*  =  ?  3-^  3<»,  3-\  etc.  Write  all  the  values  in  a 
series  from  the  exponent  +  4  to  —  4. 

(5.    Reduce    to    positive    exjwnents    6  a~^^V  h-  2  abc~^; 

M-i^-n  _j.  ,^-1.    (,3„,;,-4„  X  2--nni)  -4-  ^rOin-ni 

4 

(6.   Reduce  to  positive  exponents, 

-,   _,         1         a-^b'c^ 

(7.   Simplify 

129.  Simple  Roots.  —  Dividing  a  number  by  one  of  its 
roots  two  or  more  times  will  give  the  quotient  unity,  since 
a  root  of  a  number  is  one  of  its  t^puil  fa<*tors  (60). 


114  TEXT-BOOK   OF    ALGEBRA. 

1.   Roots  of  Monomials. 

(1.  Example.  —  Having  a  root  of  a  quantity,  say  the 
fourth  root  of  81  a^h^'^c'^,  to  find,  we  see  that  if  we  extract 
the  fourth  root  of  the  coefficient  and  then  divide  each 
exponent  by  4,  we  get  a  quantity  (?>a%^c),  which  being 
divided  into  81  a^h^-c^  and  then  into  the  quotient,  and  so 
on  four  times,  gives  a  quotient  1.  Therefore  by  the  defini- 
tion, 3  a^hh  is  the  fourth  root  of  81  a^b^'^c'^.  This  process,  it 
is  plain,  applies  generally. 

(2.    Rule  for  extracting  the  root  of  a  monomial. 

(1)  Extract  the  required  root  of  the  coefficient  as  in 
arithmetic. 

(2)  Divide  each  exponent  in  tlie  literal  part  by  the  index 
of  the  root  required  to  obtain  the  exponents  of  the  respec- 
tive letters  in  the  root. 

(3)  Find  the  sign  of  the  root  by  the  rules  in  45. 

(3.  Exercise  in  the  extraction  of  the  roots  of  mono- 
mials. 


(1)        -y/W^I^.  (5)        ^-^WW. 

(2)  ^W^iW.  (6)    ^-S2a^^ 

(3)  ^WxY-  (7)    ^6'2^pY'- 

(4)  ^'256.  (8)    ^8T^i«. 

(4.    Different  ways  of  regarding  the  same  ])Ower. 

(1)  Tlius  S  ,f^  =  (a')'  =  (ay ;  a''  =  (a")-  =  {a^  =  (a^. 

(2)  <f^  —  //'  =  the  difference  of  two  sixth  powers  of  a  and  b, 

also     =    "  "  "  the  cubes  of  a^  and  //^ 

also     =    "  "  "    "     squares       of  a'^  and  //', 

(3)  a'-'^b'  =  (a'')' -\- (by  =  (a'^-'+(b'y\   , 

(4)  </!«  -  b'''  =  (a-y  -  (by  =  (ay  -  (by  =  (ay  -  (by. 

2.    Rules  for  finding  the  square,  cube,  etc.,  roots  of  poly- 
nomials will  be  given  in  Chapter  XVIII. 


.\L(iKi;i;.\i<'   M>rAriM\  115 

130.   Theorems  in  Division :  Divisibility  of  Binomials.  — 

1.  TilKoHK.M  i.  Th<'  difffrenn'  of  the  s<tme  jtotvcrs  of  two 
f/itnnflfles  is  dii'ls'ihlc  hij  the  ditferenre  of  the  quantities. 

(1.  I.ct  the  student  verity  the  following  by  performing 
the  divisions  indicated  (124). 

(A  -  B)  ^  (A  -  B)  =  1 ;  (.V  -  j;-)  -^  (A  -  W)  =  A  -h  H. 

(A3  _  I^-')  ^  (A  -  B)  =  A-^-h  AB  +  B-^. 

( A^  -  B*)  -r-  (A  -  B)  =  A''  +  A'^B  +  AB^  +  W. 

( A*^  -  B*^)  -f-  (A  -  B)  -  A^  -h  A"'B  +  A'^B*^  +  AB='  +  B*. 

(A«  -  B«)  H-  (A  -  B)  =  k^-{-  ^'\^  +  A3]V^+  A-^B'HAB*+  B^ 

From  these  examples  we  are  led  to  suppose^  that  the 
theorem  as  given  is  true. 

(2.    Rule  for  writing  down  the  quotient. 
We  derive  this  rule  from  an  inspection  of  the  examples 

given. 

(1)  The  first  term  of  the  quotient  is  the  first  term  of  the 
dividend  divided  by  the  first  term  of  the  divisor. 

(2)  Disregarding  signs,  the  second  term  of  the  quotient 
is  found  by  dividing  the  first  term  of  the  quotient  by  the 
first  term  of  the  divisor,  and  then  multiplying  this  result 
by  the  second  term  of  the  divisor. 

{\\)  The  third  term,  of  the  quotient  is  obtained  from  the 
second  in  the  same  way  as  the  second  was  derived  from 
the  first,  and  so  on.     All  the  signs  in  the  quotient  are  plus. 

2.  Theorem  II.  The  difference  of  the  same  even 
innri'i's  of  tiro   quantities   Is    a/so   divisible   by  the   sum   of 

the    qiKUlfitit'S. 

(1.    Verify  the  following  divisions. 
(A*-^  -  B'^)  ^(A  -h  B)=  A  -  B;  (A*  -  B*)h-(A  +  B)  =  A'' 
-  A^B  -h  AB*^  -  B** ;    (A«  ~  W)  -4-  (A  +  B)  =  A*^  -  AM^  + 
A^B^  -  AW  +  AB*  -  B^ 

•  The  truth  of  the  theorems  of  division  for  any  power.-*  whatever  i»  commonly 
shown  by  mathematical  Induction  in  advanced  algebra. 


116  TEXT-BOOK   OF   ALGEBRA. 

From  these  examples  we  are  led  to  suppose  the  theorem 
true. 

(2.  The  rule  for  writing  down  the  quotient  is  the  same 
as  for  theorem  I,  with  this  exception,  the  signs  are  alter- 
nately 2:>lus  and  minus. 

3.  Theorem  III.  The  sum  of  the  same  odd  powers  of 
two  quantities  is  always  divisible  by  the  sum  of  the  quan- 
tities. 

(1.    Verify  the  following  divisions. 

(A  +  B)  -J-  (A  +  B)  =  1  ;  (A^  +  B'^)  --  (A+  B)  =  A^  - 
AB  +  B-^. 

(A^  +  B^)  -f-  (A  +  B)  =  A^  -  A^B  +  A^B-^  -  AB^  +  B^ 
(A^  4-  B^)  -^  (A  +  B)  =  A«  -  A^B  +  A^B^  -  A^B^  +  A^B* 

-  AB^  +  B«. 

These  examples  suggest  the  theorem. 

(2.    For  the  quotient  use  the  rule  given  for  theorem  II. 

Remark.  —  When  the  second  term  of  the  divisor  is  minus,  the 
process  of  division  in  these  examples  makes  every  successive  divi- 
dend plus,  so  that  eveiy  term  of  the  quotient  is  plus.  When  the  sec- 
ond term  of  the  divisor  is  plus,  only  every  other  successive  dividend  is 
plus,  and  the  signs  of  the  quotient  are  thus  made  to  alternate.  The 
process  of  division  shows  also  the  reason  for  the  rule  given  above  for 
writing  down  the  quotient.  In  actual  division  we  first  multiply  a 
quotient  term  by  the  second  term  of  the  divisor  and  afterwards 
divide  by  the  first  term  of  the  divisor.  By  the  rule  we  divide  first 
and  multiply  afterwards,  which  is  easier. 

4.  Theorem  IV.  The  sum  of  the  same  even  powers. of 
two  quantities  is  not  exactly  divisible  by  either  the  sum 
or  the  difference  of  the  quantities. 

(1.   Perform  the  following  indicated  operations  : 

(A2  +  B2)  -J-  (A  +  B)  =  A  -  B  +  /^'      i.e.,  is  not  ex- 

A  4"  jd 

actly  divisible. 


ALGEBRAIC    NOTATION.  117 

( A^  +  B^)  -V-  (A  +  B)  =  ?  (A«  4-  B«)  ^  (A  +  B)  =  ? 
^A«  4-  K«)  ^  (A  +  B)  =  ?  (A«  -f-  B«)  ^  (A^  -f  B'^)  =  ? 
(A-^  4-  B-^)  ^  (A  -  B)  =  ?  (A^  +  B^)  -^  (A  -  B)  =  ? 
(A«  +  B«)  -f-  (A  -  B)  =  ? 
(A«  4-  B«)  -r-  (A-^  4-  B-)  =  ? 

The  last  is  exactly  divisible,  because  it  really  comes 
iiider  Theorem  III.,  although  it  seems  to  belong  here. 

(2.  The  quotients  may  be  written  by  the  previous  rules, 
-f  the  proper  remainders  are  also  found  and  added. 

5.  P^xamples  in  the  theorems  of  Divisibility.  —  (With 
l>articular  reference  to  powers  which  are  themselves  simple 
powers.) 

(1.  What  is  x^  —  21)f  divisible  by?  Ans.  a:  — 3y; 
tor  it  is  the  difference  of  the  third  powers  of  x  and  ^y, 
Theorem  I.  The  quotient  is  x^  4-  3  a;  4-  9.  For  x*  -i-  x  =  x"^; 
then  to  find  the  second  term  of  the  quotient  (x"^  -^  x)  X  S 
=  '^x',  to  find  the  third  term,  (3  a;  -4-  ic)  X  3  =  9.  The 
signs  of  the  quotient  are  all  plus. 

(2.  16 1/*  —  81  is  divisible  by  what  binomial  ?  By 
theorem  1.  it  is  divisible  by  2y  —  3.  For,  -v/lfTy^  =  2  y 
and  ^Hl=3.  To  find  the  quotient:  16y* -5- 2y  =  8y«; 
(Sy»  -  2y)  X  3  =  12./;  (12 y^  -j-  2y)  X  3  =  18y;  (18y  - 
2  y)  X  3  =  27.  The  signs  are  all  4--  Hence  the  quotient 
is8y»4-12y*4-18y  +  27. 

(3.  By  what  is  x*  4-  32  y*  divisible  ?  Am.  a-  4-  2  y. 
Theorem  III.,  since  V^=a:  and  -N/32p  =  2y.  To  find 
the  quotient :  — 

x^  ^x  =  :r';  (./•*  ^  x)  X  2  y  =  2  x*i/;  (2  x^y -i- x)  X2  y  = 
4  x'Y;  (4  x'Y  -^x)  X2y  =  Sxy*;  (Sxf/*-i-x)X2y=  16  1/. 
Quotient,  x*  —  2  x^y-i-  4:  x'Y  —  Hxy^-\-  16  1/. 

(4.  What  is  m?  -  7i«  divisible  by  ?  (Of.  129,  1,  (4,  for 
this  and  the  following  examples.) 


118  TEXT-BOOK    OF    ALGEBRA. 

This  quantity  is  to  be  looked  upon  as  the  difference  of 
two  cubes,  viz.,  of  ni^  and  tv^,  and  by  theorem  I.  is  divisible 
by  m^  —  v?-. 

(m^  —  n^)  -7-  (m^  —  v?-)  =  m^  -\-  m^n'^  -\-  7i^  by  the  rule. 
(5.   What  is  x^  —  j/^  divisible  by  ?     By  Theorem  II 

(x^  —  j/^)  -j-  (x'^  +  y)  =  ^''^  —  -^V  +  ^^l/^  ~  'f  (fourth  roots.) 
{:z^  —  if)  -^  {x^  +  if)  =  x"^  —  f  (square  roots.) 

Ix'  _  y^-)  ^  (x-^  -  y)  ==  ''  (Theorem  I) 

(6.    What  is  x^  —  tf  divisible  by  ?     By  Theorem  I 
{x^  _  ^9)  ^  {x  --y)=x^^  xhj  +  ./•«,//'  +  ^V'  +  a^y  +  -^'V 

+  x'if  -h  x,/  +  //^ 
{x''  —  y^)  -H  (cc^  -  7/)  =  x"  +  x'l/^  +  ?/«. 

(7.    What  factors  has  x^  —  y^  ? 

By  Theorem  I.,  (.x«  -  ;/)  ^  (.x  -  y)  =  ?  (x^  -  ?/)  -^ 
C;^^  -  r)  =  ?     (x«  -  //«)  -  (..^  -  :/)  =  ? 

By  Theorem  IL,  (x^  —  ?/)  -^  (x  +  //)  =  ?  (x«  -  y')  -r- 
(x'  +  //^)  =  ?     (x'  -  f)  -^  (x'  +  y')  ■■=  ? 

(8.    What  are  the  factors  of   a^  —  1  ?     (See    Remark 
in  117.) 

By  Theorem  I.,  ((/«  -!«)--  (a  -  1)  =  a'  +  a'  +  rt^  _|.  ^^2 

+  a  +  l. 

(^6   _   16^    ^    (,,2  _   12-)    _  9      (,,6   _   1)    ^   (,,3 

-   1)    =  ? 

By  Theorem  II.,  (a^  -  1)  ^  (a  +  1)  =  ?  (^^'  -  1)  -^  («' 
+  1)=? 

(9.    AVhat  are  the  factors  of  x^  +  x^'  ?    One  is  x*^  +  x^ ; 
what  is  the  other  ? 

131.    Exercise  in  the  Use  of  the  Theorems  of  Divisibility.  — 

The  quotient  factors  are  to  be  written  down  directly,  by 
reference  to  the  appropriate  rule,  and  not  to  be  found  by 
division. 


AI.(il.I'.i:.\l»'     NOTATION. 


119 


1.  {a^-  fr)  --  I"  -A)  =  ? 

2.  I      •  >  -^  ( ///  ^  /' '  —  '^ 

3.  -  '  -      -^  I  ^  -  -  -  ' 

4.  (1  — //)  ^  (1  — //>  =  ^' 

5.  '(S^r'-fj;//'). 

8.  {a^'-h'). 

9.  (/'-^^  + /''')• 
10.  i»''~-/<n. 

11  Ct  „'  +  \^)^n)h^ 

12.  (U  (/"^  -  b\ 

13.  Km'  -   1. 

14.  ;//'  +  /''"'. 

15.  (SI  //'       1»;-.')  --  i^:\!/-2z). 

16.  </"  -h  **•'  "•'• 

17.  ;28  +  t\      r.y  .-  -h  /  :   nlsn  l.v  ■-  -f  ''" 

18.  (./•«  +  !)• 

19.  LTw;  y  +  «"• 

20.  ./'^  4-  •'■^'• 


ilso  by  .-:"  +  ?-•. 


120  TEXT-BOOK  OF  ALGEBRA. 


CHAPTER   X. 

FACTORING. 

132.  Factoring  in  algebra  is  the  process  of  separating  a 
quantity  into  others,  which,  multiplied  together,  will  pro- 
duce it. 

The  terms  used  in  arithmetic  to  describe  factors  are  used 
in  i)recisely  the  same  way  in  algebra. 

1.  A  (Urlsor  of  a  quantity  is  any  quantity  exactly  con- 
tained in  it. 

2.  A  multijyle  of  a  quantity  is  some  number  of  times  tlie 
quantity,  and  therefore  will  contain  the  quantity  itself 
exactly. 

3.  A.  prime  qiunitlfj/  is  divisible  only  by  itself  and  unity. 

4.  A  composite  qnantitu  is  divisible  by  other  quantities 
besides  itself  and  unity. 


SECTION  I. 

MOXOMIAI.S. 

133.   Monomial  Factors. 

1.    Monomials  standing  alone. 

(1.    Factor  30  ax^}/.     Ans.    2  .  3  .  T) .  a  .  x  .  x  .  2/  .  ^/  .  y. 

(2.    14  m%.        (3.    ^la'bc^        (4.    ^(a  +  b)\        (5.'  21 
(m  -h  7iY  {p  +  (jY- 


ALGEHItAK'    NOTATFON.  121 

2.    Monomials  apj^aring  in  polynomials. 

(1.  Separate  2  mhi -\- G  nnr  into  a  monomial  and  a 
j)olynomial.  2  mn  is  the  inonomial  tiictor,  being  contained 
in  each  term  of  the  polynomial  (124,  1).  The  other  factor 
is  m  -\-  .S  n. 

{2.       Separate  20  x^  —  45  jrt^  into  its  factors. 
(3.    12aa-'^-3/>:>-2  4-^-^. 

(4.  ory-xYi-yy. 

(5.    14 bc^j-  -  21  ^VV  +  7  //VV. 
(G.    G  hr-.r  -  IT)  Ar"^  -  :;  //-V-^. 

)i.  In  factoring;  tli«'  Hrst  thinji  to  Im^  done  with  any  jjiv«Mi  poly- 
nomial is  to  scok  for  a  monomial  far-tor  in  it,  and  if  on<'  is  found  to 
removr  it. 


SECTION    11. 

BlN<).MIAI>. 

134.    Tin*  Faitorinsj^  of  Binomials. 

1.  r/tr  (Ufference  of  iim  st/mnws  is  fdefnnil  info  fJir  sum 
II nd  difference  of  their  roots. 

This  is  the  converse  operation  of  tin*  third  th«M»r»'iii  nf 
mhitiplication 


By  the  first  theorem  of  division  9  7n^  —  U)  n-  is  divisilih* 
by  3  711  —  4  »,  and  by  the  second  it  is  divisible  by  3  m  -\- 
4n.  Or,  more  simply,  reversing  tlie  theorem  of  multijjli- 
cation  the  differenc^e  of  the  two  sqnares  is  factored  into  tlu' 
prodnct  of  the  snm  and  difference  of  the  sqnare  roots. 

9  7n'^  —  IG  )r  =  (.*i ;//  4-  4  >/ )  ('^  )n  —  4  //). 


122 


TEXT-BOOK    OF    ALGEBRA. 


(2.  a""  -  h\ 

(3.  a^U^-c'd:'. 

(4.  a^-  —. 

^  49 

(5.  <da-'-A.m\ 

(6.  4  -  .X'-. 

.r,  .,              /'- 


(9.    1-81 


(8.    lOy-^-l. 


(16.    ^x' 


(10. 

4  «■-' 

-25. 

(11. 

36  a- 

•--my\ 

(12. 

m'x 

—  30  n^x. 

(13. 

a'  - 

b\ 

(14. 

a'  - 

ah\ 

(15. 

a''"'  - 

-  b'\ 

-  25 

'/••'■ 

2.    TJie  difference  of  the  same  powers.     See  Theorem  I., 


130. 


(1.    a}  -  16. 


P)y  the  theorem,  a*  —  16  is  divisible  by  (x  —  2  and  the 
quotient  is  a^  _^  2  ti"  +  4  a  +  8.     (See  130,  1,  (2.) 


(2. 

.r-'  -  if. 

(7- 

m^n^  —  jj'K 

(3. 

x'  -  8. 

(8. 

n^  -  lA 

(4. 

r-  -  y\ 

(0. 

04  ,f  _  10 

(5. 

f  -  216. 

(10. 

a'-'  -  c\ 

(6. 

8  a'  -  27 

/A 

(11. 

,f  _  343  ',• 

3.  The  difference  of  the  same  even  powers.  Theorem  II., 
130. 

By  the  preceding  case  the  difference  of  anij  powers  is 
divisible  by  the  difference  of  the  quantities,  and  l)y  the 
present  theorem  the  difference  of  even  powers  is  divisible 
l)y  tlie  sum  oi  the  quantities.  Hence  the  difference  of  two 
even  powers  can  be  factored  in  two  ways,  and  two  sets  of 
answers  should  be  given  to  the  following.     See  130,  2,  (2. 

(1.    m^  —  n^. 

Vyy  this  theorem  m^  —  n^  is  divisible  by  ni  +  n,  and  the 
quotient  is  m^  —  nrn  -f-  ^ntv  —  w'l     By  the  preceding  theo- 


AHJKI'.KAIC    NoiAlloN.  1  J:'. 

rem  it  is  divisible  by  ///  —  n,  iind  the  quotient  is  m^  -f  nrn 

(2.    j'^  —  if.  (9.    1()./-^  —  (;25. 

(,3.    f/«  -  04.  (10.    n%'  -  rV^ 

(4.   //  -  1.  (11.   </"  -//V. 

(5.    Sl<,^_10^\  (12.    a«-  1. 

((•».    //^*-<Sl.  (13.    729-?-«. 

(7.    a;«  -a:y.  (14,   a'' —  h\ 

(8.    a-»^_?/«.  (15.    96.T*-486y. 

(10.  72  .r^  -  lir>2. 

4.  The  Sinn  of  the  Sff/tn'  ftm   odd  /joirers.     Tlieorciii  III., 

130. 

(1.    3-8  + 27. 

I>y  the  theorem  t^  +  27  is  divisible  by  x  -{■  3,  and  tlie 
(juotient  .s  .v'^  —  3  a-  -f-  9.     See  130,  3,  (2. 

(2.   ./••-[- 27//.  (9.   .r''4-04.-». 

(3.    x'  H-  .t;^  (10.    :r«  +  128  xz'. 

(4.   27  m^  +  8  w^  (11.   x^  +  //«. 

(5.  ;s'  -h  1.  (12.  a^*^  +  f,'\ 

(6.    1+04  <f«.  (13.   8  aV/V  -|-  1. 

(7.    ?//«  +  8  »^  (14.    S  .r=^  -f-  ()4  .y3. 

(8.    r^  -f-  32  .s-^  (15.    4  .r«  -I-  lOS  .-«. 
(K;.    243-^-1-  1. 

5.  liminnnils  si'jt  I I'lihlr  into  t ir  t  f rni'im mis. 
(1.    n'-\-\h\ 

This  binomial  is  not  divisible  l)y  ii  binomial.  See  theo- 
rem IV  in  division.  Nor  ean  it  be  factored  as  it  stands.  It 
will  Ih?  shown  in  136,  2,  however,  that  a  (juadrinomial  which 
is  the  difference  of  two  squares  can  be  factored.  Now  it 
is  possible  to  make  such  a  quadrinomial  of  the  given  ex- 


124  TEXT-BOOK  OF  algp:bra. 

pression.  For,  to  make  a*  -\-  4  b*  a  square,  4  a%^  must  be 
added ;  and  if  this  term  be  also  subtracted  the  expression 
will  be  unaltered.     Thus, 

a*  +  4  ^^^  =  «^  +  4  a%^  +  4  Z^^  _  4  ^2^,2^ 
which  is  a  quadrinomial  the  difference  of  two  squares.     By 
the  article  referred  to, 
(a'  _|_  4  a%^  +  4  h')  -  4  r(2/>2  ^  (^2  ^  o  ah  +  2  P)  (a'  -  2  ^/> 

Exercise  in  factoring  such  binomials  will  be  given  in 
136,  2. 

SECTIOX    111. 
Thixo.mi  AI,S. 

135.    Trinomials. 

1.  Trinoinidl  Square,^..  —  See  thecu'ems  T.  and  II.  in  mul- 
tiplication. Reversing  the  process  of  squaring  in  theorems 
1.  and  II.,  114,  we  derive  tliis  rule  for  factoring  trinomial 
squares :  Extract  the  square  I'oot  of  the  first  term  and  of 
tlie  last;  if  the  middle  term  is  twice  the  product  of  these 
two,  the  trinomial  is  a  perfect  square,  and  the  sum  or  dif- 
ferenc^e  of  tlie  square  roots  is  one  of  the  two  equal  factors 
according  as  the  middle  term  is  plus  or  minus. 

-s/x^  =  .r  :   V(>4  =  8,  2X8-  x  =  16  .r,  .-.  .r^  +  16  x 
+  ()4  =  (x  +  S)  (x  +  <S)  =  (x  +  S)-^. 
(2.    .r^  +  10  X  H-  25.  (4.    x'  +  12  xr:  -\-:M'>  z\ 

(;>>.    .r-2+  4  ic  +  4.  (5.    \)  X'  +  :^()  X  +  25. 

(C.    y^,f  +  22.r//-  +  121  .V-. 

^S.    4./-' -f  12. /••-// +  *.>.'/'. 

(1).  ;^{;./-^4-S4.'-v--f  H>  V 


AlAiKHIlAlC    .\t>i.\IH)N.  1  ll."> 

(10.   9/ +  30/ +  25. 
(11.   a:«— (3^  +  9. 

\x'^  =  a-.  V9  =  .S,  L'  X  .<•  X  .S  =  ()  .1', .-.  ;f-  —  () ./ 
-^[)  =  U-:\)(^r-3)  =  (x-'3y. 

(VJ.   a--^  —  I'C, // +  1(;9. 
(1^.    a-^_a-  +  i. 


14.   4x-^--a-//  +  ^. 
3    -^  ^  9 


9             15 
(15.   —  t/'^ //  +  25. 

(K).  y'  —  '2x+l. 

(17.  r><.^—  I()a«^;  +  5i2 

(18.  24  ic-7>  —  72  x'b^  +  54  «?.». 

(19.  «2»  — 2</"  +  1. 

(20.    m'  —  ^nr-  +4. 

2.    Simple  trhiovilals  not  squares. 

Reversing  the  theorem  of  116,  8,  we  learn  tliat  if  one 
term  of  a  trinomial  contain  as  a  factor  the  square  root  of 
another  term,  it  may  be  factored  into  two  binomials  liaving 
this  square  root  as  one  term  in  each.  The  other  terms 
must  be  so  chosen  that  their  sum  is  the  coefficient  of  tlie 
square  root  and  their  product  the  third  term. 

(1.   a-2+7x  +  12. 

The  second  term  contains  a*,  which  is  the  square  root  of 
the  first  term.  Also  the  coefficient  of  x  in  the  second  term 
equals  4  +  3,  and  the  third  term  =  4X3. 

Therefore  ««  +  7  x  +  12  =  (ar  +  4)  (x  +  3). 

Verify  this  by  actual  multiplication,  and  the  reasons  will 
become  more  clear.  Of  course  the  particular  numbers  for 
each  separate  example  have  to  be  found  by  trial. 


126 


TEXT-BOOK    OF    ALGEBRA. 


(2.   ic2  +  5  ic  +  6 
(3 

(4 
(8 


(9, 
(10 
(11 


(5.  ;^j2  _^  29  <2,v  +  100  ^2. 

z^  +  11  ,*;  +  30.  (6.  a;2  +  21  ./•  +  110. 

^2  _^  7  ^  _|_  G.  (7.  ^2  ^  13  ^/  +  12. 

^y  +  7  ^-y^  +  12. 


r-6-2  +  23  rsr^  +  90  z\ 

m^  +  17  mnj)  -\-  52  7z,^^^. 

a^  —  11a  -\-  24.     Here  the  sum  must  be  —  11,  and 


the  PRODUCT  4"  24. 

a^  _  11  a  +  24  =  {a  -  8)  {71  -  3). 

(12.   :c2  —  9  X  +  20.  (15.   x«  -  15  x^  +  54. 

(13.   a;2  —  11  X  +  30.  (16.    x'""  —  19  £c»'  +  90. 

(14.    48  _  14  :z;  +  x\     .      (17.    a%''  -  13  «/>^j  +  40  c\ 

(18.    a-2  4-  9  if  —  22.     Here  the  sum  must  be  +  9,  and 
the  PRODUCT  —  22. 

x^  _|_  9  a;  —  22  =  (a-  +  11)  {x  —  2),  since    11 
-2  =  9,  and  11  X  -  2  =  -22. 

3  m       1 
"4         4* 


(19.    a^  J^2a-  15. 


(22.    m-^  + 


(20.    x^  +  x^-  30. 


(21.    ,f  +  \y        ^ 


16- 


(23.    chl^^-cd6-  —  e\ 
^  ^3  9 

(24.    a-y  +  7  ic^^  —  8. 


(25.    ./■-  —  4. T  —  21.     Here  the   sum  must  be  —  4,  and 
the  PRoiH'CT  —  21. 

y^  _  4  .r  -  21  =  (x,  -  7)  {x  +  3),  a.p    -  7  +  3 
=  -  4,  and  -  7  X  3  =  -  21. 


(26.   x'  -X  -  30. 


(29.    a' 


(27.    7/-^  -  4  //.^  -  45  -^•^.       (30.    m'P  -  3  v/i"^^'  -  70. 
(28.    cWc"  ^  Q,  abc  —  m,    (31.    x^' -  ^yx^'i/'  -  104.  yK 


AUJKIiKAlC    NOT.vrioN.  1 -J  1 

3.     Trinomials  the  jiriKhnf  of  ninj  tint  himnuials. 

(1.  Let  us  first  take  any  two  binomials  which  will 
give  a  trinomial  product,  say  2x  -\-'^  and  .S  ^  -}-  4,  and  form 
their  product. 

'2  X  -\-  '^  '^'    i'<»^^'>    ^>  ''''  +  1^  •'■  +  1-    h  ' 

3x  -j-  4  given  to  find  its  factors,  we  know 

6  ic^  -}-  y  ^  jiist  three  things  concerning  them 

•^  %x  -\-V2  as  the  multiplication  in  the  margin 

i\x^-\-\'l  X -{-V2  shows. 

First,  that  the  product  of  the  two  coefficients  is  (5. 

Second,  that  the  product  of  the  last  two  numbers  is  12. 

Third,  that  the  sum  of  the  cross  products  is  17  x. 

Now,  if  the  product  is  given  to  find  the  factors,  we  do 
not  know  whether  the  (>  is  })roduced  by  multiplying  2  by  3, 
or  6  by  1;  or,  whether  the  12  results  from  the  multiplica- 
tion of  3  by  4,  or  2  by  0,  or  12  by  1 ;  for,  we  do  not  know 
in  advance  what  arrangement  of  the  coefficients  will,  upon 
cross  multiplication  and  addition,  give  the  17. 

Wliat  has  been  said  suggests  the  following  rule. 

(2.    Kule. 

(1).  Take  a  set  of  four  coefficients,  the  product  of  the 
first  terms  l)eing  ecpial  to  the  first  coefficient,  and  the  prod- 
uct of  the  last  terms  l)eing  erpial  to  the  last  coeffijrient,  and 
examine  whether  upon  (rross  multijilicitioii  iIkv  will  give 
the  middle  coefficient  of  the  trinomial. 

(2).  If  the  first  set  chosen  docs  not  give  the  iniildh' 
coefficient,  try  another  arrangement,  and  so  on  until  a  set 
is  found  .satisfying  the  last  condition,  or  it  is  shown  that 
none  will  answer,  which  would  indicate  that  the  trinomial 
is  a  prime  quantity. 


128  TEXT- BOOK  OF  ALGEBRA. 

(3.  Exercise. 

(1).    Required  the  factors  of  6  x^  —  25x?/  -\-  4:  y^. 
Using  2  and  3  as  the  factors  of  6,  we  write  the  following 
sets, 

2  —  2  2  —  4  2  —  1 

3  —  2  3  —  1  3  —  4 

middle  coefficient  =  —  10'   middle  coefficient  =■  —  14     middle  coefficient  =  — 11 

Next  using  6  and  1  as  the  factors  of  6 

6  —  2  6-4  6  —  1 

1  —  2  1  —  1  1  —4 

middle  cocfficent  =  —  14     middle  coefficient  =  —  10     middle  coefficient  ^=-  —  2o 

The  last  arrangement  gives  the  middle  term  as  desired. 
Hence,  ^  x^  —  'liS  xy  -\-  ^  if  =  (^  x  —  y)  {x  —  A:  y).      Ans. 

(2)  9  a;2  +  9  .^  +  2.  (11)  12  .X'-  -  31  x  -  15. 

(3)  3  ic^  +  13  x  +  14.  (12)  15  z'  -  224  z  -  15. 

(4)  4a;2_|_llif-.3.  (13)  2-ix'-2^)xy-\7/, 

(5)  9 ^.2  _^  04  ^  +  7.  (14)  'S-\-l\x^-4.x\ 

(6)  3ir'-  +  10.T//-<Sy-.  (15)  20  -  9  it' -  20  a-. 

(7)  2x^-x-\.  (16)  2x'  +  x^-2^. 

(8)  3  x'^  -  19  ;r  -  14.  (17)  24  a'^/>V  -  37  ahc  -  72. 

(9)  2  6--^  -  13  rcZ  +  6  f/-^.  (18)  6  +  32  x- -  21  .^-. 
(10)  2  tn'  -  3  my  -  2  y\  (19)  5  x'-  -  1  .r.v  —  fV  x\ 

4.  Trbiomials  of  the  form  9  a^  —  4  tt'^^^  -j-  4  ^^. 

By  adding  16  a'^h"^  to  the  second  term  and  subtracting  it 
again  in  a  fourth  term,  as  was  done  in  134,  5,  the  trinomial 
becomes  a  quadrinomial  which  is  the  difference  of  two 
squares.  Exercise  in  factoring  such  trinomials  will  be 
given  in  2  of  the  next  article. 

5.  Trinomials  the  Product  of  a  Binomial  and  Trinotnial. 
If  we  multiply  3  .r^  -j-  4  ./•  +  5  by  3  a^  —  4,  the  coefficient 

of  x^'  in  the  product  is  zero,  and  the  product  reduces  to  a 


ALGKIillAlc    NolAriON.  1«3 

trinomial.  In  the  suniu  way  o  x- -\- li  x  —  4:  ami  3a*-f  4 
multiplied  together  give  zero  as  the  eoetticient  of  the  first 
IKJvver  of  X  in  the  product.  Such  trinomials,  having  their 
coefficients  related  in  one  or  other  of  these  two  ways  may 
be  factored. 

(1.   4r'*-43x-21. 

To  find  the  factors  assume  that  2  is  the  first  coefficient  in 
eacdi  factor.  The  second  term  of  the  binomial  is  one  or  other 
of  the  factors  of  21.  Trying  7  as  the  second  coefficient  in 
each  and  3  as  the  third  term  of  the  trinomial,  the  factors 
assumed  are  2ic-  -|-  7ir  +  3  and  2  a:  —  7,  which,  upon  cross- 
multiplication,  give  —43  a;.  Hence  these  quantities  are 
the  two  factors  required. 

(2.    2oa-»-fila--12.  (4.    21  a-^  +  2«  a-' +  25. 

(3.   8«»  — 24ar'-|-2o.  (o.    \)  x^ -\- o  x -^  TAK 

SECTION   IV. 

QUADIMNOMIAI.S. 

136.   Quadrinomials. 

1.    I'he  cube  of  a  binomial. 

(1.  The  product  obtained  in  116,  4,  is  the  form  of  a 
binomial  cube,  as  «*  -|-  2  nb  -\-  h'^  is  the  form  of  a  binomial 
square.  From  the  cul)e  i)roduct  there  given,  viz.,  {a  -f  by  = 
a'-f-  ^a%  -\-  '.U/b'^  -f-  b^y  we  derive  the  rule  for  ol)taining  the 
cube  root. 

(2.   Rule. 

(1)  Extract  the  cube  root  of  the  two  leading  terms,  the 
first  and  last  as  usually  arranged. 

(2)  With  these  roots  see  if  the  middle  terms  are  respec- 
tively three  times  the  square  of  the  first  into  the  second, 
and  three  times  the  first  into  the  square  of  the  second. 


130  TEXT-BOOK    OF    ALGEBllA. 

(3)  The  sum  of  the  roots  or  their  difference  (depending 
on  whether  all  the  terms  of  the  quadrinomial  are  positive, 
or  the  second  and  fourth  minus)  is  one  of  the  three  equal 
factors  of  the  quantity. 

(3.    Exercise. 

(1)  8  a^  -  36  a'b  +  54  ab-  -  27  b\ 

-v/8^  =  2a;  -\/27l«  =  3^*;  and  3  (2  a) ^  (3 5)  =  36 a^^*, 
the  second  term  ;  3  (2  a)  (3  ^)"^  =  54  ab'^.  Since  the  second 
and  fourth  terms  are  negative  the  cube  root  is  2  a  —  3  6. 

(2)  a3_|_3^^2_^o^_^l^ 

(3)  Ub^  +  A^b^^l2b^  ^  1. 

(4)  8  ic^  -  60  x^ij  +  150  xf  -  125  y\ 

(5)  2\Q>x^ -\mx^y^\^xy^-y\ 

2.    Qiiadrino7)iials  the  difference  of  two  Squares. 
(1.  JFactor  x^  -\-  2  xy  -\-  y^  —  a^. 

Writing  the  quadrinomial  as  the  difference  of  two 
squares,  we  have  {x  +  y)'^  —  a^  which  by  134,  1,  is  factor- 
able into  the  sum  and  difference  of  its  square  roots. 

{x  +  yY  _  a'^  =  (  (x  +  y)  +  a)  {{x^y)  -  a)  =  {x^y 
+  a)  (x  f  //  -  a). 

(2.    4  a^  -  (9  h"-  -^bc^  c"). 
(3.    4a2_(9^2_  12  be -^ic^). 
(4.    4:a'--9b''-\-c''-\-4:ac. 

In  this  and  some  of  the  following  examples,  the  order  of 
the  terms  is  disarranged,  and  to  unite  the  terms  properly 
is  the  first  thing  to  be  done.  Here  the  first,  third,  and  last 
terms  go  together  to  make  the  square  of  2a-\-c.  Hence 
we  have  (2a-\-  ey  —  9  b'^,  the  factors  of  which  are  (2a-\-  c 
■i-Sb)  (2a-^G-3b). 

(5.  P  —  m'^  —  n^  -\-  2  m.n 
(6.  2ab-\-a''  -x''  +  b.^ 
(7.    b^-l-2ab  +  a\ 


AIXiEBUAIC    NOTATION.  131 

(8.    (Sa^-^by-  (a2-;UV 
(9.    10  -  O^a  -  2/0^. 

(10.  x*-y'  i-vjj-     .;<; 

(11.  -4ir-\-x'-\-'J//r:~\iz\ 

(12.  42  aO  -1-1-41)  a-  -  \)  //-. 

(13.  -V2af,--\a--\-i)y'-'Jb\ 

(14.  2ab-a'-b'-\-  1. 

(15.  a^  -f  2  be  -  ^-^  -  6=^. 

(l(>.  y""  -(i/-  z)-^". 

(17.    »'  +  .,-y/^-f/A     (S(H'  135,  4.) 

Suggestion.  —  Add  and  subtract  a-lr.     Thus  we  get 

{n*  +  2  a-'^^^i  +  6*)  -  a^/^  =  {n^  +  '/-  +  ch)  {,r^  +  /,^  -  rr/;). 

(18.   25x<-36a-«y*-h4//'. 

SuOGESTlo.N. — To    have    ;i    1 1  iiiuinial    square,    the    inichUe    term 
ought   to   be  4- 20  i-^//- or  —  20  J-?/"-     If  —  10  xV"   *>e   taken   out   of 

—  30jr'V"  a"tJ  a  fourth  term  b«'  made  of  it,  tlie  expression  can  be 
factored  by  this  case. 

(25  I*  -  20  xV  +  4  y^)  -  10  x-^y-'  ^  (5  x^  -  2  //-^  +  4  xy)  (5  x-^  -  2  y2 

-  4  xy). 

(19.  ir>x^-i7ry +  //. 

(20.  9ir*  +  38j:y +  49./. 

(21.  9a*  +  21aV  +  25c*. 

(22.  25  X*  -  41  a;2 1/  +  16  /. 

,'_';;.  .,•"  -|-r,4.     (See  134.  ."».") 

SicjGESTioN.  —  A(hl  and  subtract  10 x-^.      Thus,  x^+ 10x^  +  04 
-16x-^. 

(24.   4a;*  +  81«^ 
(25.    64xy  +  81«\ 
(26.    4  m*  -h  ()25. 

3.    Quadrinomials  the  product  of  binomials. 

(1.    Examples. 
(1)    Factor  (tr  -f  od  -\-  he  4-  hd. 


lo2  TEXT-BOOK  OF  ALGEBRA. 

Here  a  can  be  taken  out  of  the  first  two  terms  and  h  out 
of  the  last  two,  giving 

which  is  plainly  divisible  by  the  binomial  c-\-d^ 
c  +  d)  a{c-\-d)-\-  h  (c  +  d) 
a  -{-b 

Therefore  the  factors  of  ac  -\-  ad  -\- he -{-  bd  are  c  -\-  d  and 
a-\-h. 

(2.    Factor  Qax  —  2  by -{■  ?,hx  —  ^  aij. 
Taking  3  x  out  of  the  first  and  third  terms,  and  2  y  out  of 
the  second  and  fourth 

Q  ax  -  2by  +  ^bx  -  4.ay  =  ^x  {2  a  ^h)  -2y  (b  +2  a) 
2 a  +  b)  :ix(2a  +  h)-2y{b^2a) 
Zx-2y 
.'.    6ax-2by-\-3bx  —  4:ay  =  (2a-\-b)(3x-2y), 

(2.    Rule. 

(1)  Take  a  monomial  factor  out  of  two  terms,  and  a 
second  monomial  factor  (if  necessary)  out  of  the  two 
remaining  terms.  If  this  is  done  properly,  a  binomial  fac- 
tor is  seen  directly  (i.e.,  if  the  quadrinomial  can  be  factored 
at  all). 

(2)  The  other  factor  is  found  by  division. 
(3.    Exercise. 

(1)  ac  —  bd  -\-  be  —  ad. 

(2)  a^-{-a^-\-a-{-l. 

(3)  jcfj  -3y^2x-(j. 

(4)  Q  ax  -  I  ay- 21  xy-\-Uy'. 

(5)  9  am  — 4: bm  —  27  an  -\-\2bn. 

(6)  m,^  —  jnn  -\-  m^n^  —  n^. 

(7)  x^  —  xy^  —  x^y  -{-  7f. 

(8)  cdx'^  —  cxy  -\-  dxy  —  y\ 

(9)  x^  —  xhj  —  xjf  +  y^. 


ALGEIJUAIC   NOTATION.  133 

(10)  abcy  —  h^dy  —  acilx  -\-  hdhc. 

(11)  12x»-8j-//  — 9a;y +  Gy«. 

(12)  15^^«-J-12«-H-10«-|-8. 

4.    Qiiadrinomials  tiie  product  of  binomials  and  trinomials. 

(1)  4  a;2  _  9  y2  _2  a-«  -f  3  yz. 

Factoring  the  first  two  terms  and  taking  z  out  of  the  last 

t\v.> 

2  a;-  3  y)  (2  a;  -f-  3//)  (2x  -  3//)  -  ;g  (2  a;  -  3y) 
2x-f3y-« 

(2)  a;»  -  6  a;2  -f  G  a;  -  1. 

a;-l)a;»-l-6a;(x-l) 

x«-|-a;  +  l-6a;  =  a;2_5x+l. 
.*.  the  factors  are  x  —  1  and  x^  —  5  a;  4- 1. 

(3)  m»  +  5/»2  +  5  7/?.  -h  1. 

(4)  4  a%'  -  1(>1)  c'  4-  6  aZ»(/  +  39  cd. 

(5)  *♦  10j;»  +  aj*^-:z;-28. 

Assuming  that  this  can  be  factored  into  a  binomial  and  trinomial, 
let  us  set  down  trial  coefficients,  leaving  the  middle  term  of  the 
trinomial  blank. 
^    .J  _  /  V     _  -  Choosing  as  coeflficients,  5,  —  7,  2,  and 

2  -  4-4  4,  as  indicated,  and  multiplying  ox^  by  4 

iQ  j^  J-  /  w^  —  14  3;  *^^  product  is  20  x^.    Since  the  coefficient 

20x*             —  28    of  x^  in  the  product  is  1,  the  product  of  2 
10x8                           —  28    t)y  the  blank  coefficient  of  x  must  be  —  19. 
But  this  value  makes  the  coefficient  of  ar, 
—  52  instead  of    —  1.      Hence   this  ar- 
K^  _7  rangement  of  the  coefficients  fails. 

■xci'T*  4.  t\'\   -2  4-  *>n  '1^\\\»  arrangement  gives  —  14  j*"^,  which 

1  \^ ^2  -^\)x  -  28    ^q"''^»  +  0^">)  J^'^  I.e.,  the  coefficient  +  3 
10  x"*  +  x'^ —  —  X ^^^^28    ^^  ^^^^  trinomial.     Now  +  3  in  the  tri- 
nomial gives   —  dC  in    the    product    as 
desired.     Hence,  2  x*  +  3  ac  4-  4,  and  5  x  —  7  are  the  factors  sought. 

(6)  4x«-21a-2  +  44a-  — 30. 

(7)  6a;»-3a-2-33a--  6. 


134  TEXT-BOOK   OF   ALGKBllA. 

SECTION   V. 
Polynomials  of  more  than  Four  Terms. 

137.   Factoring  Polynomials  of  more  than  Four  Terms. 

1.  Expressions  which  are  powers. 

(1.  A^  +  4  A«B  +  6  A'-^IV^  +  4  AB^  +  V>\ 
If  an  expression  of  five  terms  have  two  fourth  powers 
and  three  other  terms  formed  out  of  the  two  roots  as  the 
middle  terms  above  are  formed  from  A  and  B,  it  is  a  per- 
fect fourth  power,  and  can  be  factored  into  an  expression 
of  the  form,  (A  +  B)^  or  (A  —  B)^ 

(2.    16  x^  —  96  x^y  +  216  x^'if-  -  216  x?/  +  81 7/. 

VT6x'  =  2x',   ^/Sly'  =-^37/;  now,  4  X  (2  x)^ 

X  —  3  y  =  —  96  x^ ;     6  X  (2  xy  X  (-  3  j/Y 

=  216  xY ;     4  X  (2  x)  (-  3  yY  =  -  216  xi/. 

Therefore  the  factors  are  (2  x  —  3  i/y. 

Note.  —  Higher   powers   than   the  fourth  contain  more 

terms,  and  involve  a  greater  number  of  conditions,  but  are 

solved  in  the  same  way  as  the  third  and  fourth  powers. 

2.  Polynomials  the  difference  of  two  squares. 

(1.    Factor  a^  _^  2  ah  -\- U' —  c'' -\- 2  cd  —  d}  =  (a  +  hy 
—  {c  —  dy. 

The  factors  are  the  sum  and  difference  of  the  roots  a  -\-h 
and  e  —  d.     (Cf.  134,  1,  and  136,  2). 

a  -{-h  -{-  e  —  d  and  a  -\-h  —  c  —  d  =^ 
or,  a  -\-h  -\-  c  —  d  and  a  -\-h  —  e  -\-  d. 

(2.    («  +  ^>  +  cy  -d\ 

(3.    4  ^2  _  12  m/i  +  9  71^  —  ^)2  _|_  4^^  _  4  ^2^ 
(4    {^a^h-^cy-ie^f^gy. 
(5.    {a^h^c-\-dy-{e-fy. 
(6.    (3  x'  —  4  X  —  2)2  -f  (3  x-^  +  4  X  —  2)2. 
(7.    4  {ah  +  edy  -  (a^-  4-  IP-  —  (^  ~  d'^Y. 


AL(;i;i!i;.\ic  N(>'r.\'rM»v.  135 

3.  Polynomial  the  square  of  a  trinomial. 

(1.    a^  +  4  b'  4-  9  6-=^  -h  4  ai  +  6  af  4-  12  be. 
Fixtnicting  the  roots  of  the  square  terms,  we  have  a,  2  6, 
and  3c.     The  other  terms  are  double  the  products  of  these, 
and  therefore  (See  116,  3),  the  factors  are  {a  -\-2b  ■\-  i^  r).'^ 

(2.   a.^  -h  9 1/'  +  25  c2  —  6  ab  -f-  10  ac  -  SObc. 

4.  Othef  Polijnomiah. 

(1.   x^  +  L>  x>j  -f  ,/  +  C.  ./•  -h  ('.  //  .»!•,  {..'  +  ///-  -f  <>  (.^  +  y) 
wliicli  is  evidently  divisible  by  (jc  -}-  //). 

^  +  y)  («_+j^)'  +  6  fe+_y) . 

(2.    *  6  x*-^  —  11  a^y  -h  3  //-^  -  u-^  —  7  //,i;  -  2  ;?2 

Factoring  ^Ae  ,/zVs^  three  terms  as  in  135,  3,  and  setting 
down  tlie  factors  in  the  customary  way,  we  have, 

2a.-3y, 
^x  —  y. 

We  see  now  that  if  -|-  5;  be  annexed  to  the  first  factor 
and  —  2 «  to  the  second,  we  sliall  obtain  the  additional 
terms  of  the  expression. 

Tlierefore  2  x  —  3//  -|-  ,~  and  .»  x  —  //  —  2  z  arc  the  fac- 
tors sought. 

(3.   x'i -2xy-\-y'^^  5x  —  5y. 
^  (4.   2x^-xy-3y^-5yz-2z*. 

(5.   2a^'\-(jax  —  18  «  4.  4  x*  —  24  x  4-  36. 
(6.    2a^  —  4ab-4  ae  -f  2  b'  -|-  4  /yr  +  2  e\ 

_j_  lei)   _^  (;  ,^/,,. 

This  is  the  form  of  the  cube  of  a  trinomial,  as  may  be 
verified  by  forming  and  arranging  the  iircxhict  (a  4-  ^^  +  ^) 
(a  +  ^»  4-  c)  (a  4-  ^  +  ^). 

(8.    «»  —  ^»  —  r«  —  3  {a%  —  ab''  -\-  a^c  —  ac'  -\-  b'^c  -\-  be') 
4-  6  a^c. 


186  TEXT-BOOK  OF  ALGEBRA. 

General  Remark.  —  When  the  number  of  terms  in  a  poly- 
nomial exceeds  four,  it  is  usually  difficult  to  factor  it. 
And,  as  a  rule,  the  greater  the  number  of  terms  the  greater 
the  difficulty  of  factoring. 

In  the  foregoing  classification  of  algebraic  expressions 
with  respect  to  the  number  of  their  terms  it  is  intended 
that  the  expressions  are  to  he  in  their  eximnded  forms.  Other- 
wise there  will  be  more  or  less  confusion.  Thus  {a  +  ^)^ 
—  c^  is  a  quadrinomial,  not  a  trinomial  or  binomial. 

138.    Promiscuous  Exercise  in  Factoring. 

1.  a^  _|_  2  aH)''  +  h\  (2.    1)  y'  -  49  a""}/. 

3.  18  x''  +  33  axij  +  14  ah/\ 

4.  16  ay'h^c^  +  24  alA-^  +  9  h^n\ 

5.  8  c"  -i\cd-:)  d\ 

6.  (1^  —  .r«. 

7.  .,3/>3  _|_  512. 

8.  a'  —  if'^  —  2yz  —  .-2. 

9.  (a  +  hy  -  r\ 

10.  8  a^  -\-  6  ah/  —  9  n ,/'  -  L>7  /A 

11.  54  a^inx  +  12  ahn^  -f-  IS  aui. 

12.  (7  X  +  4  7/)2  -  (2  X  4-  3//)2. 

13.  x^  —  6  x'^t/  4-  12  x>r  —  8  /A 

14.  6  a?^  —  5  .T?/  —  6  ?/"^. 

15.  8  «2  +  2  rj  -  3. 

16.  ?/i^x'  +  m^i/  —  ?iV  —  7?^//. 

17.  1  —  a^x^  —  //y  +  2  r//>x;/. 

18.  ah/  -  />V'  -  ^''^^//'  +  hMx\. 

19.  »-^  +  hx^  -\-  ax  -{-  ah. 

20.  .//^/^  -  r^'^  -  Z;2  +  1. 

21.  rr*  —  7  a--  ~  18. 


AI  <;iJii;AI('    NOTATION. 


187 


22. 
23. 
24. 
25. 
26. 
27. 
28. 
29. 
30. 
31. 
32. 
33. 
34. 
35. 
36. 
37. 
38. 
39. 
40. 
41. 
42. 
43. 
44. 
45. 
46. 
47. 
48. 
49. 
50. 


x(x-{-z)  -  1/  (//  -h  z). 

^'  +  //^  +  '^  ^if  (-^  +  y)' 

a%c  —  acSi  —  (ifnl  -\-  hcj-. 

in^-^   n'^  -\- p'^  +  li  mn  -\-  2  mp  -\-  '1  np. 

c^d*  _  c^  _  a^»d^  -I-  a'\ 

(a  +  t,y'  -h  (a  +  b). 

a'-'  -  /A 


aVr 


1 


18  x^t/z  +  y  a-=^//-^,t  +  ()  x\f/z^  H-  3  .r}/'z\ 

()  am  -\-  4  an  -\-  \)  bm  -\-  i\bn. 

1  _  (r  _  yY- 

a^  -  (b  -  r)\ 

(a  _3:r)2-16//2. 

S(r +  //)«-  (2  .>•-//)». 

a^  -  /A«. 

„4  _|_  0.%-^  -  //V  -  r*. 

:r^  —  7  a*V*  +  ?/• 

31  J-  _  3;-)  -  6  x\ 

ax  Of  +  A")  +  />//  (^''•^^+   ^W 

1  4-  (A  _  ,,.2)  j.-J  _  ,thx\ 

3  ««  _  6  ^/>  -j-  3  //^  +  6  ^r  —  <•)  A/'. 

(a-^-f  //-  +  2  <^^»)-  -  (^'^  -h  //-^  -  2  ///>)'^. 

(^'  +  y'  +  v'  -  -'  ^//  +  2  .r.tr  -  2  //-)  -  0/  4-  ^y 

z^+{x-,jy-2z{x-y). 

m*  —  w*  —  m  (m^  —  n^)  +  n  (m  —  w)*-^. 

(a-^-^a  -4)- -4. 

J.-  _  (J.  _  6)2. 


138  TEXT-BOOK    OF  ALGEBKA. 


CHAPTER   XL 

COMMON    FACTORS. 

139.    Common  Factors.  —  Highest  Common  Factor. 

1.  A  Coiunioii  Factor  of  two  or  more  quantities  is  a  fac- 
tor that  appears  in  each  of  them. 

2.  The  Highest  Common  Factor  of  two  or  more  quanti- 
ties is  the  product  of  all  the  prime  factors  common  to  each. 

a.  A  common  factor  is  the  same  as  a  common  divisor.  The  high- 
est common  factor  (abbreviated  into  h.  c.  f.)  in  algebra  corresponds 
to  the  greatest  common  divisor  in  aritlimetic.  Indeed,  by  many 
authors  the  latter  is  the  term  used  in  algebra. 


SECTION  I.    First  Method. 
Th?:  Highest  Common   Factor  by  FACTORiNa. 

140.    Principle  involved  in  finding  the  h.  c.  f .  by  Factoring.  — 

If  we  have  a  quantity  expressed  as  the  product  of  its  fac- 
tors, each  factor  is  a  divisor  of  it,  and  furthermore,  the 
product  of  (iny  set  of  its  lyrime  facfors  is  also  a  divisor. 

Thus,  the  prime  factors  of  2?A0  are  2,  8,  5,  7,  and  11. 
Now  any  one  of  these,  of  course,  is  a  divisor  of  2310 ;  the 
product  of  any  two  of  them,  as  .33  (=  11  X  3),  is  likewise  a 
divisor ;  so,  also,  the  product  of  any  three  of  them  as  70 
(=2X5X7);  and  so  on.  And  finally  the  product  of  all 
of  the  factors  is  a  divisor,  being  the  number  itself. 


AL(Ji:i;i;ai<'  notation.  1B9 

Similarly  the  product  of  any  set  ot  t\w  i)riine  fiictors  of 
a^  —  i^,  (ly  X  —  I/,  X  -^  //,  ./•■-  -f-  //-,  and  x*  -f  t/*)  is  a  divisor 
ofa:*  —  /. 

It  is  evident  from  what  has  just  hceu  said  that  the  prod- 
uct of  all  factors  common  to  two  or  more  quantities,  (i.e., 
found  in  each  of  them)  is  a  divisor  of  each  of  them.  More- 
over, this  is  the  h.  c.  f.,  since  if  an  additional  factor  were 
introduced  all  of  the  quantities  would  not  contain  the  result. 

If,  then,  we  have  the  different  quantities  exi)ressed  as 
the  ])roduct  of  their  factors,  we  can  pick  out,  merely  by 
inspection,  all  those  which  are  common,  and  so  have  the 
factors  of  the  h.  c.  f. 

To  illustrate.  Suppose  it  is  required  to  find  the  h.  c.  f. 
of  a*  —  2ab  -\-  b'^,  a'^  —  P,  and  ar  —  he 

Factoring  each  quantity, 

«2  _  2ab  +  b'^  =  {a  —  b)  (a  -  b)  , 
a^-b^  =  (a  +  b)  (a  -  b). 
ac  —  be  =:  c(a  —  b). 

Here  a  —  ^  is  a  divisor  of  each  product,  and  therefore  by 
definition  the  common  factor. 

To  illustrate  still  further,  let  us  find  the  h.  c.  f.  of  12  nV> 
(a  ~  xy  (a  +  xf,  3  a%  {a  -  xf  (a  +  ./•)«,  and  9  a^b'  (a  -  xf 
{a  -I-  x). 

12  a%  (a  -  xy  (a  -f  x)*  =  12  a%  (a  -  x)  (a  -  x)  (a  +  x) 

(a^x)(a-\-x). 
3  a^b  (a  -  xy  (a  -f  xy  =  8  a%  (a  -  x)  (a  -^  x)  (a  +  x) 

(a -{- X)  (a -^  x). 
9  a*b''  (a  -  xy  (a-{.x)=  9  a'b''  (a  -  x)  (a  -  x)  (a  -  x) 

(a+x). 

By  inspection,  we  see  that  3a^b  is  the  product  of  all  the 
monomial  factors  common ;  (a  —  xy  is  the  highest  power 
of  a  —  ar,  and  (a -{- x)  the  highest  power  of  a -j-o;,  wliich 
are  contained  in  each  of  the  quantities.  Therefore,  by  defi- 
nition, the  h.  c.  f.  is  3  a^b  (a  —  x)'^  (a  -\-  x) 


140  TEXT-BOOK   OF    ALGEBRA. 

141.  Rule  to  find  the  h.  c.  f .  of  two  or  more  Quantities  by- 
Factoring. 

1.  Separate  each  quantity  into  its  prime  factors. 

2.  Choose  out  the  greatest  coefficient  and  the  highest 
power  of  every  other  factor  that  will  still  be  contained  in 
each  quantity. 

3.  The  product  of  these  will  be  the  h.  c.  f. 

142.  Exercise.  —  Represent  the  h.  c.  f .  as  the  product  of 
all  the  prime  common  factors. 

1.  6  a%,  9  a%,  24  a^'xij.        2.    284  a,  126  a,  210  a. 

3.  12  ab\  and  25  Z/V. 

4.  48.ry;^^  12  xhfz%  24:xyz\  20x'ifz\ 

5.  2a^-2  ab%  and  U  {a  +  bf. 

6.  35  a^2)xy,  and  42  b\xy. 

7.  12  a^xhj  -  4  a^xif,  and  30  a^xhf  -  10  a^xhf. 

8.  .T^  —  ?/2,  x^  —  ?/^,  and  x'^  —  1  xy  -\-^  y'^. 

9.  a-^  _  2  J!  -  3,  and  0^2  _  ^  _l_  ^2. 

10.  3a^8_^6iz;2_24ic,  and6cc^-96:r. 

11.  3:^2-6^  +  3,  6a:2  +  6aj-12,  andl2a'2_l2. 

12.  ^6  _  ^8  _  30  .p6  _  ;i3  ^3  _^  42^  and  ir«  +  x^  -  42. 

13.  ic^'"  +  o!'"  —  30,  and  ic^'"  _  x'^  —  42. 

14.  «c  (a  —  b)  (a  —  c),  and  ^c  (^  —  a)  (b  —  c). 

Note,  b  —  a  =  —  (a  —  b),  and  a  —  ^  is  contained  in 
(b  —  a),  —  1  times. 

15.  a^b^  -  4  a'b',  and  a«^2  _  iq  ^2^6^ 

16.  ««  +  3  a^^  +  2  ab%  and  a^  +  6  a»6  +  8  a^^,^. 

17.  a^  —  a^o;,  a^  —  ax^,  a^  —  a^r^ 

18.  x^  —  xy^,  and  x^  -\-  xSj  +  o-y  +  t/^. 

19.  a:;3  4-  3  ic^y  _|_  2  ic?/2,  ic^  +  G  .t2?/  +  5  x?/^. 


AI.(;i;i5KAIC    NOIA'PION.  141 

20.  3a^  —  'iab-\-0',^a'-6a''0-{-a'0\ 

21.  2h'  -  5h  -^2,  rjb"^  -Sb-'  -3b  -\-2. 

22.  x'  -  1,  x'  -  2x  -  3,  iSx^  -X  -  20. 

23.  (ja^^lax  —  3  x\  (S  n^  +  11  au-  +  3  x\ 

24.  3x2-|-16ar-35,  5x^4-33.r-14. 

25.  ch:^  —  d^f  acx"^  —  bcx  -\-  adx  —  bd. 

26.  2x2  H-  9a;  +  4,  2x2  +  liar  +  5^  gx^  -  3x  -  2. 

27.  3x^  +  8x«  +  4x2,  3x« -h  11  x^ -I- 0  x»,  3./-^  -  lGx« 
-12x2 

28.  8a»  +  l,  16a^4-4a2_|.l. 

29.  8x«- 27,  16x^  +  36x2  +  81. 

Kf:m  AKK.  —  The  method  by  factoring  is  not  adapted  to  the  sohition 
of  problems  whose  expressions  are  difticult  to  resolve  into  their  fac- 
tors; and  recourse  is  then  had  to  tlir  method  by  continiu'd  divisioij. 


SECTION    11. 

FlM»IN<.    TIIK    H.    V.    V.    \\\    CONTIM  KI>    DIVISION. 

143.  Principles  involved  in  finding  the  h.  c.  f.  by  Continued 
Division. 

1.  The  definitions  and  |ninciples  of  the  first  method. 

2.  A  divisor  of  a  nnniber  (or  quantity)  is  a  divisor  of 
any  nund)er  of  times  that  number  (or  quantity.) 

If,  for  example,  5  is  contained  in  20,  it  is  contained  in 
{'^)  (=z  '.\  X  20)  three  times  as  often  a,s  in  20.  It  is  contained 
in  20  four  times,  and  in  (>()  tliree  times  four  times,  or  twelve 
times. 

If  ft  is  contained  in  J.  b  times  exactly,  it  is  containe(l  in 
2  J.  2b  times  exactly,  and  in  ///.I.  /ttb  times  exactly. 

3.  A  common  divis(»r  of  two  numhers  (^or  (quantities;  is  a 
divisor  of  tlnMr  sum  or  difference. 


142                           TEXT-BOOK    OF   ALGEBRA. 
This  follows  from  124,  1. 
Thus,  ^)  24 +  ^JL6 ^_9^   . 

4  times    -[-  .11    times    =15    times 

11   times  2    times    =       9  times 

144.    Application  of  the  Principles  to  justify  the  Method  of 
finding  the  h.c.  f.  by  Continued  Division. 

1.    Application  to  an  arithmetical  example.  —  To  find  the 
g.  c.  d.  of  258  and  731. 

Explanation.  —  731  is  divided  by 
258  and  contains  it  twice  with  a  re-     258")  731  {9 
mainder  215.     Then  258  is   divided  515 

by  this  remainder,  and  so  on.     43  is  215)  258  (1 

contained  in  215  exactly  and  is  the  215 

g.  c.  d.     It  is  required  to  prove  that  43)215(5 

this  process  gives  the  g.  c.  d.  of  the  215 

two  numbers. 

(1.  Is  any  divisor  of  258  a  divisor  of  516  ?  By  which 
principle  ?  Is  any  common  divisor  of  516  and  731  con- 
tained in  215  ?  By  which  principle  ?  Does  it  follow  that 
any  common  divisor  of  258  and  731  is  a  divisor  of  215  ? 
Why?  Can  it  be  inferred  from  this  reasoning  that  the 
g.  c.  d.  of  258  and  731  cannot  he  greater  than  215  ? 

(2.  Furthermore,  is  any  common  divisor  of  215  and 
258  (and,  of  course,  of  twice  258)  also  contained  in  731  ? 
State  the  principle.  Would  it  be  allowable  then  to  drop 
731  entirely  and  proceed  as  before  with  215  aad  258  ? 
Also,  after  a  second  similar  operation  with  43  and  215  ? 

(3.  Is  43  the  g.  c.  d.  of  itself  and  215?  By  the  fore- 
going reasoning  is  43  contained  in  258  and  731,  and  can 
their  g.  c.  d.  be  greater  than  43  ?  Does  it  follow  then  that 
43  is  the  g.  c.  d.  sought  ? 


ai.(;i:i;i:ai<'    \<  m  a  i  m  -n.  1  j;*, 

By  tlie  preceding  lUftliod  we  always  nave  two  num- 
bers before  us  from  which  the  h.  c.  f.  required  is  to  Ih^ 
fouiul.  First,  the  two  numbers  themselves  are  taken  ;  then 
tlie  divisor  and  a  third  number  derived  from  the  first  two; 
then  the  last  derived  and  a  fourtli,  and  so  on.  Now,  every 
derived  number  used  was  obtiiined  by  principles  2  and  />'. 
But  these  j)rinciples  will  give  other  numl)ers.  Thus,  a 
divisor  of  l^'iS  and  I'M  is  a  divisor  of  their  sum  *.)89,  or  their 
difference,  473;  and  25<S  and  989,  or,  258  and  47:>,  may  1)  • 
Uoed  instead  of  258  and  731. 

2.  Algebraic  demonstration  of  the  metliod  of  finding  the 
li.  c.  f.  by  continued  division.^ 

To  find  the  h.  c.  f.  of  A  and  B. 

Dividing  as  in  the  margin,  just  as  in  the  arithmetical 
example,  letting  the  Q's  and  R's  stand  for  the  successive 
(juotients  and  remainders,  sup^)ose  R2  is  contained  in  Rj, 
Qa  times  exactly ;  then  Rg  is  the  h.  c.  f.  sought.  The  i)roof 
follows  the  lines  of  that  for  the  arithmetical  problem. 


A)  B  (Q 
QA 

• 

B  -  QA  = 

H)  A  (Q, 
Q.R 

A  -  Q,R  =  R,)  R  (Q, 

R  -  Q2H1  =  R2)  Ri  (Qg 

Ri— Q8K2=o 

(1.  Every  divisor  of  A  is  also  a  divisor  of  QA  (prin.  2) ; 
every  common  divisor  of  I^  and  QA  is  a  divisor  of  iXw'n 
difference,  R  (prin.  3)  ;  hence,  any  common  divisor  of  B  and 
A  is  a  divisor  of  R,  and  so,  of  course,  cannot  he  higher  than 
a.  Furthermore,  any  common  divisor  of  R  and  A  is  also 
a  divisor  of  B  Cprin.  3);  therefore  B  can  be  dropped,  and 

<  Tilts  drnionstratioii  .«liutild  be-  omitted  by  beg^innera. 


144  TEXT-BOOK    OF    ALGEBRA. 

the  process  can  be  continued  with  A  and  R ;  in  the  next 
operation,  the  common  divisor  of  A  and  R  cannot  be  higher 
than  Bi,  and  so  on,  as  long  as  it  is  necessary  to  continue 
the  operation.  Hence,  the  h.  c.  f.  cannot  be  of  a  higher 
degree  than  the  last  divisor. 

(2.  The  last  divisor  is  contained  in  the  given  quanti- 
ties, as  also  in  each  of  the  remainders,  as  is  made  evident 
by  the  factored  values,  (140). 

By  hypothesis  Rj  contains  K2,  Qs  times.     Therefore 

II  =  Q2H1   +  R2  =  Q2  (QsRs)   +  1^2  =  (Q2Q3  +  1)^2  (43) 

A  =  QiR  +  R,  =  Q,  (Q2Q3  +  1)  R2  +  Q3R2  =  (Q1Q2Q3  +  Qi 

+  Q3)K2 

B  =  QA  +  R  =  Q  (Q1Q2Q3  +  Qi  +  Qs)  B2  +  (Q2Q3  +  1)  B2 

=  (QQ1Q2Q3  ^  QQi  +  QQ3  +  Q2Q3  +  1)  K2 

145.  Method  and  Principles  in  the  Solution  of  Algebraic 
Examples.  —  Modifications.  The  argument  by  which  we 
have  proved  that  the  process  of  continued  division  gives 
the  g.  c.  d.  of  two  arithmetical  numbers  applies  equally  well 
(as  we  have  just  seen)  in  algebra.  But  there  are  a  number 
of  very  important  modifications  in  algebra.  To  show  how 
these  arise,  and  how  they  are  dealt  with,  an  algebraic  ex- 
ample is  given. 

It  is  re(pured  to  find  the  li.  c.  f.  of 

S./.5  _  K)./.^  __  ;5(l./-'5  -^  lS.r-.  and  IS./-*  —  m,,-'^  —  (k-  -f  12x. 

Taking  out  tlie  monomial  factors 

8^.5  _  \{\jA  _  ;5(|.;.J  _^  is.r-'  =  '2x-  {A.r'  -  S./-'^  -  ITu-  +  9) 
18.r^  —  (\{)y^  -  {\y'  +  72j-  =  iSx  (}\x^  —  \i)x'  -  X  +  12). 

In  this  form  we  see  by  inspection  that  2x  is  the  monomial 
factor  common,  and  therefore  a  factor  of  the  h.  c.  f.  Since 
we  have  no  read}'  nunins  of  factoring  the  polynomials   to 


ALGEBRAIC   NOTATION.  14o 

see  if  they  have  a  cominon   la*  t    i.  we  resort  to  division  as 
in  144  to  tind  their  h.  c.  i". 

IJut  we  are  confronted  at  the  outset  with  the  dihMimia 
that  whichever  polynomial  is  made  the  dividend,  the  first 
term  of  the  other  will  not  be  contained  in  it  exactly. 
Thus,  4x^  is  not  contiiined  in  3x*,  nor  vice  versa.  To  obviate 
this  dithculty,  and  others  of  a  similar  character,  certain 
coursf^s  are  pursued  in  algebra  which  were  not  necessary  in 
arithmetic;.    They  depend  upon  the  principles  already  given. 

1.  IVe  maf/j  if  we  choosej  multiply  one  of  the  two  quanti- 
ties in  hand  by  a  factor  not  found  in  the  other. 

2.  We  may  disrard  a  factor  found  ifi  one  quantity  and 
not  in  the  other. 

For  710  cowjnon  factor  is  either  introdured  into,  or  taken 
aivay  from  the  h.  c.  f.  However,  we  mast  not  multiply  one 
quantity  by  a  factor  found  in  the  other ;  for  in  so  doing  we 
make  this  factor  common  to  the  two,  and  introduce  an 
irrelevant  factor  into  the  h.  c.  f. 

3.  It  is  immaterial  at  any  time  whether  a  quantity  itself 
is  used,  or  its  opposite.  The  h.  c.  f.  when  found  may  have 
one  value  or  its  opposite.  For,  finding  a  h.  c.  f.  is  simply 
a  question  of  divisibility. 

To  give  a  simple  illustration,  if  6  is  a  divisor  of  -f-  30,  it 
is  containeil  in  —  .*)()  also;  and  if  —  3  is  contained  in  18,  so 
;.l>.Ms  +  :;. 

a.  In  order  to  change  the  sign  of  a  quantity,  we  must  change  the 
sign  of  each  term  (100,  2,  4).  To  change  the  sign  of  a  polynomial 
in  the  present  case.  It  is  customary  to  divide  through  hy  a  negative 
factor  if  any  is  to  come  out,  otherwise  by  —  1.  It  is  preferable  to 
retain  the  first  term  of  polynomial  quantities  positive,  changing 
signs  wlien  necessary. 

Let  us  i)roceed  now  with  the  example  begun  above  by 
first  multiplying  ^or*  —  li)r^  —  ar  +  12  by  4  so  as  to  make  it 
divisible  by  4x*  —  <Sr^  —  IT)./-  -f  \). 


2d 
quo. 


146  TEXT-BOOK   OF    ALGEBRA.    ~ 

3^3  _  lOct-2  _  it;  +  12 

4 

12x^  _  4{)x-^  -  4cc  +  48       (4^3  _  <5,^2  -jL5^j4-_9 
12a;3  -  24a;^  -  45x  +  27         (_3   ist  quo. 

_  1  (  -  16x^  -\-  ilx  +  21  (4)  

df4.   16x2  -  Ux  -  21)     IGa-'^  -  '.V2x'  -  6i)x  -\-  36  (; 
16^3_  4ia;2_21ic 

3  (  9x-^  -  39^  +  36 
J^,,  Wx'  -  Ux  -  21  (3x'  -  i:ix  +  12  a^,_ 
15ic"^  —  6ox  4-  60  (5  3d  quo. 

x^  4_  24i^  _  8l  "~ 

3  ^2  -  13  X  +  12  (a;2  +  24;r  — 81 
3a;2  +  72a;  — 243  (3.  4th  quo. 
-  85  (-  85  ;r  +  255 

a;  — 3)  ic'^  +  24  ^  -  81  (x  +  27 
ic^  —  3j? 

27  a; -81 

27  a;  — 81 

Hence  a?  —  3  is  the  h.  c.  f.  of  the  two  polynomials,  and 
2x  (x  —  3)  the  h.  c.  f .  of  the  original  quantities. 

Explanation.  —  The  first  divisor  is  multiplied  by  4  in  order  to 
contain  the  first  remainder.  The  second  remainder  is  divided  by  3 
following  the  directions  in  2.  of  this  article.  There  is  danger  if 
this  were  not  done  of  introducing  the  factor  3  into  the  other  polyno- 
mial, and  .3  would  thus  become  a  common  factor,  which  would  be 
manifestly  wrong,  as  neither  of  the  polynomials  we  began  with  con- 
tains 3.  Instead  of  multiplying  3  a:- —  13  a;  +  12  by  16,  or  10  x'-  — 
41  X  —  21  by  3  to  make  the  first  term  of  the  dividend  contain  the  first 
term  of  the  divisor,  and  in  so  doing  obtaining  large  coefficients,  we 
divide  10  x-  —  41  j-  —  21  by  3  x-  —  lo  x  +  12  obtaining  a  remainder  of 
the  same  degree  as  the  divisor.  This  amounts  to  nuiUiplying  3  .r-  — 
13  X  +  12  by  5  (prin.  2),  an.l  subtracting  the  product  from  10  .r-  — 
41  X  —  21  (prin.  3).  The  remainder  is  used  with  3  x'^  —  13  x  +  12  to 
continue  the  operation  (144).  The  divisor  a;  —  3  is  contained  in 
its  dividend,  and  is  therefore  the  h.  c.  f.  of  the  polynomials. 


ALGEBRAIC   NOTATION.  147 

146.   Rule  for  finding  the  h.  c.  f.  by  Continued  Division. 

This  method  may  be  used  to  sohc  every  kind  of  })roblem, 
but  it  sliould  only  be  used  in  the  case  of  polynomials  too 
difficult  to  factor.  It  is  assumed  that  all  monomial  factors 
have  been  taken  out,  and  either  retained  for  the  h.  c.  f.  or 
suppressed  as  no  longer  needed. 

1.  According  to  the  rule  for  division  126,  3  divide  the 
jx)lynomial  of  higher  degree  by  the  other  (or  if  of  the  same 
degree  either  may  be  divided  by  the  other)  until  the  re- 
mainder is  of  as  low  or  of  a  hnver  degree  than  the  divisor. 

2.  Divide  the  divisor  by  tin  ivmainder  just  obtained  in 
the  same  way,  and  so  continut  . 

3.  That  divisor  which  is  (M.iitaiiu'd  in  its  dividend  witli- 
out  a  remainder  is  the  h.  c.  f.  sought. 

4.  If  there  are  three  or  more  quantities,  first  use  the  two 
whose  h.  c.  f.  is  most  easily  obtainable  ;  then  take  the  li.  c.  f. 
of  these  along  with  a  third  quantity,  and  obtain  their 
h.  c.  f. ;  and  so  on.  Tlie  last  one  found  will  be  the  one 
required,  for  it  will  be  the  highest  factor  contained  in  each 
j)olynomial. 

<i.  A  factor  found  in  one  quantity  and  not  in  the  other  slioiild  he 
<lis(;ird«Ml  at  any  time. 

/;.  Wi"  may  intnxhice  at  any  time  into  one  a  factor  which  is  not 
found  in  the  other  quantity. 

c.  We  may  at  any  time  multiply  one  polynomial  by  one  factor  and 
tlie  other  by  another  factor  (by  1  in  145)  and  add  or  subtract  the 
l>rotiucts  (prin.  li),  and  use  this  result  with  either  of  the  given 
polynomials,  or  with  a  second  result  similarly  obtained  to  continue 
the  operation.  This  plan  may  b;*  followed  when  the  regidar  process 
is  about  to  give  large  numerical  coefficients. 

f/.  When  a  remainder  is  obtained  which  does  not  contain  the 
letter  of  arrangement,  there  is  no  common  divisor. 


148 


TEXT-BOOK   OF   ALGEBIIA. 


147.    Examples  in  finding  the  h.  c.  f.  by  Continued  Division. 

1.    Given  x'  —  6x-\-S,  and  4  x""  —  21  x'  -f  1 5  .x  +  20. 


4.r^-21.r=^  +  15ic  +  2() 
4:X^  —  24.x'-\-32x 


4:X-\-H 


Sx^  —  lSx -\-24: 

h.  c.  f.     x  —  4.jx''  —  (yx-\-H(x--2 
x'^  —  4  X 
~^Yx  +  8 

—  2a; +  8 

2.    Given  7  x^-\-  2ox  —  12,  and  17  x^  +  67  ic  -  4. 

The  solution  of  the  problem  in  this  form  promises  to  give 
large  numerical  coefficients.  AVe  prefer  therefore  to  derive 
two  other  expressions  (146,  c). 


7  x^  -{-'2ox  —  12 

Ux^-\-50x  —  24: 
7ic2  +  2oic-12 


17  X-  +  67  X  —  4 
14  a^-^  4-00  a-  — 24 

3  a:;^  -j-  17  a?  -j~  -^^     l**'  derived  expression. 

17:/-^+  67  »•  —  4 

9 


3oa;"^  +  125a;  — 60        34  ic^  _|.  134  a;  _  8 
34a:^  +  134a-  — 8 

X'^ ^  X  —  52      2d  derived  expression. 


We  proceed  with  the  two  derived  expressions  as  if  they 
were  the  quantities  given.  The  following  arrangement  of 
the  work  of  tinding  the  h.  c.  f.  avoids  the  necessity  of  re- 
writing divisors. 


2d  quotient. 

ic  — 13 


1st  d  visor  and 
2d  dividend. 

£c2_9^_52 

x^  -\-  Ax 
-13X  —  52 
-13.x -52 


1st  dividend. 


3.^2  + 17  a; +  20 
3a-2  — 27  a;  — 156 
44(44a;  +  176 
h.  c.  f .   a;  +  4 


Ist  quotient. 

3 


ALGEBRAIC    NOTATION.  149 

148.    Exercise  in  finding  Highest   Common   Factors.  —  The 

choict'  of  method  in  left  to  the  student.     (See  reiiiiirk  in 
142.; 

1.  21)45  and  3441.  2.    (;;i(>.  IKii,  KWO. 

3.  ;J094,  4420,  2G52,  4(>G2. 

4.  ;i  y-  -i-Wx  —  So,  and  5  r^  +  3.'?  ./•  -  1 4. 

5.  2x'^-xt/-6i/%andSx^-Sxi/-\-^l/-. 

6.  6a^-{-7ax  —  3x^,  and  Sa-^  +  11  aa- +  Ga^. 

7.  « V  —  4  a^rm  -{-  3  ncm'\  and  a*c^  —  (>  a^c'^ni  -j-  5  r-m^. 

8.  6  a-«  —  7<^a-2  —  20  <^ V,  ;ind  :\  ./•-  +  7  o.r  -\-  4  //-. 

9.  2x*  —  7x*  4-  oa-^  and  ar*  -j-  Sj-  —  4.#'. 

10.  12x*  — 51a-«  +  12a-,  and  2a-*  —  4a-^  —  2a'»  +  4ar2. 

11.  3y«-13y2  +  23y-21,  and  6y» +  /  — 44y  +  21. 

12.  ^/ V»  —  a-/>r-^y  +  <///V//-  —  hYr  2  a^^^a^^y  —  ab^f/'^  —  hh/\ 

13.  2 a-=^  4-  9 a:  +  4.  2 ./-  +  11 .'  -h  T),  and  2 .r-  —  3 .r  —  2. 

14.  3 a-*  +  S  .r8  4-  4  ./•-,  ;{ ./•*  -f-  1 1  x'  +  0  .r\  and  3  a'"*  -  1  i\  r' 

—  \2x\ 

16.    .r«  —  9 ar^ -f  2C) a- —  24,  .r«  —  lO.r-' 4- 31  .r  —  30.  and  .r^ 

—  na-^4-3()a-  — 3(>. 

16.  J-*  —  2  a'\r^  -\-  a\    and    r*  4- 2  r/.r«  4- </V- —  ./- —  2  ^/.r 

—  n^  (137.  2). 

17.  4  a'i  _  r> .///  4-  h\  3  ^/«  -  3  a^h  4-  ^///-  —  /A  and  o'  —  h\ 

18.  a'«4-7a;2  — a-  — 7.    .r^  4- 5  .r-^  —  ./•  —  T),   and    (.r-^  — 2x 
4-  l)'^  (136,  3). 

19.  Ga-^  — 2r);r  4- 14.    4  r' -  20.r  +  21.     2 a-"^  -  1  r> .r  4- 2S. 
and  2x'»  4- 5a- -42. 

20.  2  a-*  —  4  .r»  4-  S  ./••-  —  1  L\/'  4-  (1.    and    .">  .r<  —  3  ,r^  —  (J  .r- 
4-9.r  — 3. 

21.  a-^  4-  ^/a-"  —  1)^/ V-  4-  11  r/«a-  —  4a^  and  ar^  —  aa-«  —  3^/ V 
4-  5  «V  —  2  a*. 


150  TEXT-BOOK    OF    ALGEBUa. 

22.  3x^-^17x-i-20,3indx^—9x  — 52. 

The  following  example  and  exercises  should  be  passed 
over  until  the  subject  is  reviewed. 

23.  Given  6  a^  +  13  a'd  —  73  ah'  +  60  b^,  and  8  a^  —  26  a% 
-^21ah-'  —  9h\ 

We  shall  use  the  first,  and  the  difference  of  the  first  and 
second  instead  of  the  two  quantities  themselves. 

Isl  Divisor.  1st  D.vidend. 

2a3-39rt26  +  100«62-r)9  63  6«3+    iSa^ft-   73a62+   60^3        3, 

05 'ZA.  Dividend    6  a^  —  11?  asj,  +  .'^(jo  ab^  —  207  b^       1st 


a 

2d 

quo. 

130  rt^ - 

130  «i- 

-23  6jj- 

2535  ii-b  +  0500^62  —  4485  b^ 
'6T.Ufib+    20~rtf>2 
2 162  «26  +  C233  ab'-  -  4485  6^' 
94^2    -   271  «6  +    195  62 
3 

l^) 

1 30  (CH)  -  .373  a62  +  267  63 
130  a2   -373«6  +207  62 

quo. 

260  a2  -740a6  +53462 

282  ar- 
200  a2 

— 

813  «6  +   585  62 
746  a6  +   534  62 

22  a2 

— 

67rt6  +     6162 

132  a2 
130  a2 

3 

402  a6  +   30662 
373^6  +    267  62 

a- 13 
4tli 

4th  Dividend 

.        2  rt2 

2a2 

— 

29  «6  +     39  62 
3«6 

22a2- 
22  a2- 

-  67rt6+  5162 
-319a6  +  429  62 

3d 
quo. 

quo. 

~ 

20  a6  +     39  62 
26  a6  +     3962 

120  6^ 

252  a6- 378  62 

2a     —       3  6      4cl,  D.Tisor. 

ll.  c.  f. 

Explanation.  —  Having  found  the  second  remainder,  —  2162a-6 
+  C233  ah''  —  4485  />'',  and  the  numbers  being  very  large,  by  tlie  method 
of  g.  c.  d.  in  arithmetic  their  greatest  common  divisor  was  found  to 
be  23.  Using  94  a^  -  271  ah  +  195  />^  and  180  a'  -  373  ah  +  267  h\  we 
derive  22  or  —  (57  ah  +  51  &"^  by  146,  c.  Next,  using  22  a'  —  67  ah  + 
51?/-  in  a  similar  manner  2  a^  —  29  rt6  +  39 //-  is  derived.  Finally, 
using  2  rt-  —  29  ah  +  39  6"^  and  22  or  —  67  ah  +  51  //-  the  h.  c.  f.  is 
obtained  in  the  usual  manner. 

24.  6:r»  +  13a-2  +  15£c  — 25,  and  2x»  +  4:^^ -f  4a:  —  10. 

25.  Q>x^  —  x^  —  3x''-Ax  —  4,  and  ^x^  —  2x^  _  ig^c^  +  3x 
+  10. 

26.  7/  -57/2  +  117/  -15,     7/^-7/2  +  3^  +  5,     2l/-7  7/ 

+  16// -15. 

27.  2  if  -\-2i/  —  3  if  —  5if-Uf  —  7  ?/,  and  2if-\-2f 
—  r)f-ryf-7y-'7. 

28.  .T«  +  4.T^-3.x'^- 16.t3  +  11.t2  +  12.t -9  and  6x^ 
+  20£c4  -  12 ic^  _  48x2  +  22x  +  12. 


ALGKBltAlC    NUTATION.  lol 


CHAPTER   XII. 

COMMON    MULTirLES. 

149.    Common  Multiples  —  Lowest  Common  Multiple. 

1.  A  multiple  of  a  qiuintity  is  literally  the  (juautity  mul- 
tiplied by  something. 

2.  A  common  multiple  of  two  or  more  quantities  is  a 
quantity  that  is  some  numl)er  of  times  each  one. 

3.  The  lowest  common  multiple  of  two  or  more  quanti- 
ties is  that  one  which  is  of  the  lowest  degree. 

a.  \  multiple  of  a  quantity  is  often  defined  as  one  that  will 
contain  it;  and  a  common  multiple  of  two  or  more  quantities  as  a 
(luantity  that  will  contain  each.  The  lowest  common  multiple 
(ahhrevlated  into  1.  c.  m.)  contains  no  factors  except  those  necessary 
to  make  it  a  multi])lc  of  the  several  quantities. 

Of  conunon  nniltiples  there  can  be  any  number,  but  there  can  he 
only  tnir  lowest  common  multiple.  To  illustrate  this,  if  we  take  any 
quantity  as  a  +  m,  then  '.\  {a  +  m),  ({ n  4-  <J  in,  n-  -\-  am,  etc.,  are  all 
nniltiples  ;  and  there  may  be  any  number  of  such  multiples.  For, 
anything  at  all  by  which  we  multiply  a  4-  m  gives  a  multiple  of  it. 
The  expression  U  «;/<■*  is  a  multiple  of  3,  a,  fin,  m,  aw,  iiani,  //^, 
etc.,  and  so  a  common  multiple  of  any  two  or  more  of  them.  It  is 
not  a  multiple  of  ni'\  nor  of  j*,  which  is  a  factor  not  found  in  It. 
\ow  Ham-  is  the  I.  c.  m^  of  the  quantities  enumerated  above.  It  is 
not,  however,  the  I.  c.  m.  of  a,  'Ha,  an<l  Hm,  for '-lam  will  contain 
each  of  them,  and  is  of  a  lower  <legree  than  H  am'-. 

h.  The  I.  c.  m.  is  used  in  adding  and  subtracting;  fractions,  in 
clearing  equations  of  fractions,  etc.  And  it  is  highly  desirable  that 
the  student  shouhl  be  thoroughly  conversant  with  the  examples  and 
exercises  about  to  b<'  <:iv«Mi. 


152  TEXT-BOOK  OF  ALGEBRA. 

150.  Principles  applicable  in  finding  Common  Multiples. 

1.  Any  number  of  times  a  given  multiple  of  a  quantity 
is  also  a  multiple  of  that  quantity.     This  is  self-evident. 

2.  The  product  of  any  number  of  quantities  will  always 
be  one  of  their  common  multiples,  since  it  will  contain  each 
of  them. 

3.  The  lowest  common  multiple  of  two  or  more  quanti- 
ties must  contain  all  the  prime  factors  of  each  of  them,  and 
no  others,  each  factor  appearing  with  the  highest  exponent 
which  it  has  in  any  one  quantity. 

For,  the  product  of  all  such  factors  is  necessary  and  suf- 
ficient to  contain  every  quantity. 

151.  Rules  for  finding  the  1.  c.  m.  of  two  or  more  Quantities. 

1.  After  factoring  each  quantity,  by  inspection  w^rite 
down  in  a  product  all  the  prime  factors,  using  no  factor 
oftener  than  it  occurs  in  any  one  quantity. 

2.  Or,  set  down  the  factored  quantities  on  a  horizontal 
line,  and  divide  by  any  prime  factor  that  will  divide  two  or 
more  of  them,  and  bring  down  the  undivided  quantities  to 
the  line  beneath.  Divide  this  new  line  of  quantities  by 
any  prime  factor  that  will  divide  two  or  more  of  them  with- 
out a  remainder,  and  so  continue  to  divide  until  the  last 
line  of  quotients  and  undivided  numbers  are  prime  to  each 
other. 

Multiply  the  divisors  and  last  line  of  undivided  quanti- 
ties together  for  the  1.  c.  m. 

A  little  reflection  will  show  that  this  accomplishes  what 
was  required  in  3.  of  the  last  article. 

3.  Method  when  the  quantities  cannot  be  factored. 

(1.  When  there  are  but  two  quantities  find  their  h.  c.  f. ; 
then  divide  each  by  the  h.  c.  f ;  and  last  of  all  multiply 
the  two  quotients  and  the  h.  c.  f.  together.  This  process 
fulfils  the  requirements  in  principle  t]  of  150. 


ALGEBRAIC   NOTATION.  153 

(2.  AYhen  there  are  more  than  two  quantities  find  tlie 
1.  c.  m.  of  two  of  them,  then  of  this  result  and  a  third,  and 
so  on.     The  last  1.  c.  ni.  is  the  one  recjuired. 

a.  There  is  usually  an  advantage  in  retaining  all  expressions  in 
their  factored  form. 

b.  When  quantities  have  no  common  factor  their  product  is 
their  1.  c.  m. 

152.   Examples  in  finding  Lowest  Common  Multiples. 
1.    Find  four  coninion  multiples  and  the  1.  e.  ni.  of  a^, 
fth^c%  and  be*. 

(1.    Their  product  a^b*c^  is  a  common  multiple. 
(2.    aV>V,  or  any  higher  exponents. 
(3.    aWc^  still  contains  each. 
(4.    aV/r*  the  1.  c.  m. 

2.  Find  the  1.  c.  in.  of  '.)  n  (x^  —  1),  (yab  (x^  —  x),  and 
4  fc^b  (x  —  \).  Factoring  the  (|uantities  and  writing  the  re- 
sults side  by  side, 

;j  a  {X-  —  1)  =  3  a  (x  4-  1)  (j-  —  1) 
6  ab  (x^  —  x)  =  2-3  abx  (x  —  \) 
4  .rV;  (j-^  -  1)  =  2-2  a%  (x  --i)  (x'  -\-  x  +  1). 
.-.  the  I.e.  ni,l)y  rule  1,  is  12rrYy.r  (x  +  1)  (x  -  1)  (x'  +  ;r  -f-  1). 

3.  Find  the  1.  c.  m.  of  x''  —  irx^  -|-  U  j.  —  6,  x"'  -  9  x^  -|- 
•jC,  .r  _  L'4,  and  ar*  -  8ar*  +  19  ar  -  1 2. 

3^  _  9  j-2  -f  2r>  X  -  24  (ar'^-Gx^-hlla;— 6 
7^-(Sx^-\-nx-iS         (1 

-  ^  (-33-^ -f  ir)x-  18 

xjj-_5x  -h  6)a:^  -(\x^ -\-\\  x  -  (S  {x- 1 
ar'  —  o.r'-  -j-  ^>  ^ 

—  x^  -\-  r»  .r  —  (5 

—  x^-^r^x  ~() 

...  ar-  —  5  a-  -f-  6  (=  (a-  —  2)  (a-  —  3))  is  the  1.  p.  m.  of  these 
two  quantities.    Dividinj?  the  third  quantity,  .r*"  —  S  a-*  -|-  19  a- 

—  12  by  J--—  5  a- 4- r».  tlicir  h.  <•.  f.   is   soon    found   to   l)c 


154  TEXT-BOOK   OF    ALGEBRA. 

X  —  3,  We  are  now  in  position  to  factor  the  three  quanti- 
ties, since  we  know  two  of  the  factors  of  the  first  two 
quantities,  and  one  factor  of  the  third.     We  find 

x^-6x^-{-llx~  6  =  (x-l)(x-2)(x -3) 
x^-9x'-\-2(Jx-24:  =  (x-  2)  (x  -  3)  (x  -  4) 
x^  -  S  x^  -\-19  X  -  12  =  (x  -  ^)  (x  -  3)  (x  -  4). 

By  inspection,  we  have  for  the  1.  c.  ni.  (x  —  1)  (x  —  2) 
(X  -  3)  (X  -  4). 

153.    Exercise  in  finding  the  1.  c.  m.  of  Sets  of  Quantities. 

1.  54,  81,  24,  27. 

2.  60,  12,  120,  48,  3G. 

3.  432,  270. 

4.  18ax2,  Taif,  12xij. 

5.  x^  and  ax  -\-  x'^. 

•    6.    x'^  —  1  and  x'^  —  x. 

7.  3a%  4.b%  2cH,  9ad^ 

8.  x^  —  y^,  x  -{-  y,  and  x  —  y. 

9.  ah'^c^x^,  a%G^x^,  and  cv'U^cx. 

10.  3x-yz^,  ir)xy^z%  10  xhj'^z^. 

11.  3  a'hc,  27  a^h'',  and  (>. 

12.  9  a'b%  12  b\  a^c^,  36,  8. 

13.  14a2^,2^  7^»V,  3,  2,  5. 

14.  2x(x  —y),4.  xy  (x^  —  y^),  6  xy^  (x  +  ?/). 

15.  4  (1  +  x),  8  (ic  -  1),  1  -  ;r2. 

16.  a'^  +  «Z»,  ai»  +  b\ 

17.  3x2,  4ic2_|_g^ 

18.  Ax^y  —  y,  2x^  -\-x. 

19.  x*^  (x  -  yY,  if  {x  4-  yy,  xy  {xP-  -  y^) 

20.  Sa''b-\-%ab%  Qa-Q>b. 


Ai.(ii:ni:Aic   noia  ri<>N, 


loo 


21.  a  (x  —  b)  (x  —  c),  b  (/•  —  jc)  {x  —  a),    ami    c  (a  —  x) 
(b  —  X).     (See  .Sug.  Ex.  14,  142.) 

22.  X'  4-  2 X,  x'^  +  3x  +  L\ 

23.  '^  {X  -it  y\'i  {X  -  y)rS{x'  ■\-  f). 

24.  a  -f  b,  a^  -\-  2  ab  +  b'^,  and  «*  -  b\ 

25.  a--  -f  5  X  -j-  0,  and  x'^  •\-  (Sx  -\-  8. 

26.  a--^  -f-  1 1  J-  4-  30,  and  x'^  -\.\2x-\-  35. 

27.  a'^  —  X',  a^  —  2  ax  -\-  x'-,  a^  -\-  2  ax  +  x^. 

28.  x'  -l,x^-\-x-\-l. 

29.  Ox^-x- 1,  2a:'^-3ic-2. 

30.  {x  -  x^,  x'^  —  1,  and  4  x  (1  +  x). 

31.  x2  -  4  «''^,  (x  +  2  ay,  and  (x  -  2  af. 

32.  x2  —  1,  x3  +  1,  x»  -  1,  and  x«  +  1. 

33.  3  (««  -  ^«),    4  («  -  b)\    5  («<-  ^^),    <;  (^/-  -  ^2)^    and 

34.  x^  -  2  x^  +  1,  X*  H-  4  ar»  +  6  x2  +  4  X  +  1. 

36.   3  x^  +  2G  x«  -f-  35  x*,   6  x-^  +  3S  x  -  2S,   and    27  x» 
H-27x'^  — 30x. 
36.    12  X-  -  23  xy  +  10  if,  4  x'-^  _  9  xy  +  5  y-,  and  3  x^  — 

2n.     X*  +  ffX"'  4-  a^x  -f-  O^  X^  +  «2^2  _|_  ^^4^ 

38.  15x3- 14x'V-f  24xy2-7y'',   j^,^j    27x»  +  33x*V- 
20x^=^  +  2^. 

39.  x2--3x?/-  lU7/-,x2-|-2xy-35//^x-  -  Sx//  +  15//-, 
and  x^ -\- A  xy  —  2\  tf, 

40.  .f'^-f-  7x  +  9,  x*-^  -  3x  -h  7,  and  x^  -  2x  +  11. 


156  TfiXT-BOOK   OF   ALGEBRA. 


CHAPTER   XIII. 

FRACTIONS. 

154.  A  Fraction  in  Algebra  is  any  expression  written  in 
the  fractional  form,  that  is,  with  a  numerator  and  denom- 
inator,  and  used  to  indicate  a  division. 

«.  Thus,  if  a  and  h  represent  whole  numbers  |  represents  a  simple 
fraction  in  algebra.  The  unit  is  divided  into  h  parts  and  a  parts  are 
taken.  Or,  if  we  look  upon  the  fraction  as  an  indicated  division,  a 
is  divided  by  />,  just  as  3  is  divided  by  5  in  %. 

But  letters  are  supposed  to  have  any  values,  integral  or  fractional. 
Now,  if,  for  example,  «  =  ^  and  b  =  f,  we  cannot  any  longer  say 
that  the  unit  is  divided  into  f  parts  and  i  part  is  taken,  for  such 
language  has  no  meaning;  but  we  can  say  that  {>  is  divided  by  |  giv- 
ing an  arithmetical  complex  fraction.  For  this  reason  it  is  custom- 
ary to  look  upon  all  algebraic  fractions  as  indicated  divisions  (Cf.  63, 
r/),  the  numerator  being  the  dividend,  the  denominator  the  divisor, 
and  the  value  of  the  fraction  the  quotient.' 

As  in  arithmetic,  the  numerator  and  denominator  are  called  the 
terms  of  the  fraction.  Besides  such  expressions  as  the  one  given, 
I ,  the  fraction  may  have  any  complex  quantity  for  either  or  both  of 

its  terms,  e.g.,  ^>  «^  +  9  qc  +  lo  6^c  ^^^  ^  trinomial  for  its  numerator 

16  ahc  +  9  c2 
and  a  binomial  for  its  denominator. 

h.  It  is  the  form  of  an  indicated  division,  and  not  any  nufherical 
value  it  may  have,  which  makes  an  expression  fractional.  Indeed, 
what  we  should  call  an  integral  quantity,  as  3  a&,  may  have  a  frac- 
tional value ;  thus,  putting  a  =  |,  ?>  =  4,  3  ab  =  |.  While,  on  the  other 
hand  ~  which  is  an  algebraic  fraction  becomes  equal  to  2,  an  integer. 

J  For  a  thoroughgoing  treatment  of  all  the  fundamental  questions  of  algebr.a 
the  teacher  should  consult  treatises  on  the  subject  like  Peacock's  or  Chrystal's. 


ALGEBRAIC   NOTATION.  157 


SECTION    I. 

('I.A>SIFI(ATI<)\     AM)    1'IM\«   IIM.K<. 

155.   Classification  of  Fractions. 

1.    With  respect  to  their  origin. 

(1.  A  Simple  Fraction,  the  original  form  of  the  frac- 
tion, contains  entire  (quantities  for  its  numerator  and 
denominator. 

Thus,    p^,    J^±^''+'\. 
5  nrn      2  nif/  -{-  3  nr:  —  f- 

(2.  A  Complex  Fraction  arises  upon  dividing  one  frac- 
tion by  another.  A  complex  fraction  has  a  fractional  ex- 
pression in  one  or  both  of  its  terms. 


nf  —  71' 

")  „/, 

in 

III  -f  7 

7'S 

Thus, 

(3.  A  Mixed  Quantity  is  one  which  is  partly  integral 
and  partly  fractional. 

E.g.,  r,„+£ 

a 
2.    With   res[>ect  to  their  capability  of    reduction  to  a 
mixed  quantity. 

(1.  When  the  numerator  does  not  contain  the  denomi- 
nator an  entire  numl)er  of  times,  the  expression  may  be 
called  by  analogy  a  proper  fraction. 

,,         iSnhi'     3r/  4-2// 
mn       \-\-  i  c 
(2.    When  the  numerator  does  contain  tiie  denominator 
an  entire  number  of  times,  it  may  be  called  an  improper 
fraction. 

E. fr,  A«_+  -i ^'   =3  4-       ^"^  (126;. 

^'   04.64-c  ^  a-l-6-f-c  ^ 


158  TEXT-BOOK    OF    ALGEBRA 

156.  Fundamental  Principle  in  Fractions.  —  If  a  fraction  is 
regarded  as  an  expressed  division,  then  multiplying  or 
dividing  both  terms  by  the  same  number  will  not  change 
the  quotient,  i.e.,  the  value  of  the  fraction. 

A  proof  of  this  principle  may  be  given  as  follows :  — 

Let  —  denote  any  fraction,  and  x  its  value  ;  then  x  =  — . 
b  ^  h 

Whenc^e.  a  =  bx  (def.,  43) 

Let  in  be  any  number;  then  from  the  equation  just  writ- 
ten, it  follows  self-evidently  that 

ma  =  mhx  (207,  o) 

or,  ma  =  inh  '  x  (38,  2) 

Therefore,  -'^  =  x.  '  (def.  43) 

7nb 

i.e,  ^'  =  «  Q.Ji.n 

mb        b 

It  follows,  conversely,  that  ])oth  terms  of  a  fraction  may 
be  divided  by  tlie  same  quantity  without  altering  its  value. 

157.  The  Three  Signs  connected  with  every  fraction.  There 
are  three  signs  expressed  or  understood  belonging  to  every 
fraction,  viz.,  those  of  the  numerator,  denominator,  and 
fraction  itself. 

The  same  is  true  of  a  polynomial  numerator  or  denom- 
inator. 

To  illustrate.    .  <^  - ''' - '\    =+      +/"' 7 '"  ~  "']  , 
—  ab  -\-  ac  —  be  -\-  (—  ab  -{-  ac  —  be) 

^    ,        +0^-f,'-e^ 

—  (ab  —  ac  -\-  be) 

parentheses  being  used  to  constitute  the  polynomial  one 
quantity. 


ALGEBRAIC    NOTATION.  loO 

The  sign  before  the  fraction  applies  to  the  valtie  of  the 
frnctlon,  i.e.,  tlie  quotient  of  the  numerator  divided  by  the 
(h^noniinator. 

1.  p]ffect  upon  the  fraction  produced  by  changing  these 
signs. 

(1.  Evidently,  changing  the  sign  l)efore  a  fraction 
changes  its  calue  from  plus  to  minus,  or  from  minus  to  ])lus. 

(2.  Changing  the  sign  of  eitli«*r  numerator  or  denom- 
inator (dianges  the  sign  of  the  quotient,  (43)  wliich  is  the 
value  of  the  fraction. 

Thus,  —  =  -h  o,  while  — - —  =  —  5,  and =  —  5. 

4  4  —  4 

(3.  If  the  sign  of  eitlier  numerator  or  denominator  and 
at  the  same  time  the  sign  of  the  fraction  be  changed,  tlie 
fraction  is  changed  back  to  its  former  sign  and  remains 
unaltered. 

(4.  If  the  signs  of  both  terms  of  a  fraction  be  changed, 
the  value  of  the  fra(^tion  remains  unaltered. 

rp.         15      .        ,  -15      K     — 18        18  o 

Tims.  —  =  ;).  and =  5 ; = =  —  6. 

:;  -3  ()  -(> 

(.">.  Finally,  since  changing  the  signs  of  the  two  terms 
leaves  the  fraction  the  same,  changing  all  three  signs 
changes  the  sign  of  the  fraction. 

2.  Ivules  for  changing  the  signs  of  a  fraction. 

(1.  Changing  one  or  all  three,  i.e.,  an  odd  numl)er  of 
the  signs  of  a  fraction,  changes  the  sign  of  the  fraction. 

(2.  Changing  any  two  of  the  signs  of  a  fraction  does 
not  alter  its  ^^alue. 


160  TEXT-BOOK   OF   ALGEBRA. 

SECTION  II. 
Keduction. 

158.  Reduction  of  Fractions.  —  Keduction  in  all  mathe- 
matics is  the  process  of  changing  the  form  of  a  quantity 
without  altering  its  value. 

a.  There  are  five  cases  of  reduction  of  fractions  commonly  given: 
reduction  to  lowest  terms;  reduction  of  an  improper  fraction  to  a 
mixed  quantity;  reduction  of  a  mixed  quantity  to  an  improper  frac- 
tion; reduction  of  an  entire  quantity  or  a  fraction  to  the  form  of  a 
fraction  having  a  given  denominator;  reduction  of  two  or  more 
fractions  to  equivalent  fractions  having  a  least  common,  or  any 
connnon  denominator. 

I. -TO   LOWEST  TERMS. 

159.  Reduction  of  a  Fraction  to  its  Lowest  Terms.  Prin- 
ciple and  Kule.  —  A  fraction  is  in  its  lowest  terms  when  its 
numerator  and  denominator  are  })rime  to  each  other. 

By  156,  dividing  both  terms  of  a  fraction  by  the  same 
quantity  changes  its  form,  but  does  not  alter  its  value. 
Therefore  we  may, 

1.  Factor  the  numerator  and  denominator  into  their 
prime  factors,  and  then  cancel  out  of  both  terms  the  factors 
common. 

2.  Or,  find  by  continued  division  the  h.  c,  f.  of  the  terms, 
and  divide  both  terms  by  it.  The  resulting  fraction  is  in 
its  lowest  terms. 

160.  Exercise  in  reducing  Fractions  to  their  Lowest  Terms. 

1. .     Dividing  both  terms  by  3  ah^c  we  get  — . 

9  aire  3  h 

Ans. 


12. 


ALGEBRA IC    NOTATION.  161 


15     jn       243  n^-n%-^ 

45'  144'   1(52  n'-b' 


,     1274     18(>07  ,^  x^-b^ 


2002'  24587  '  y' -\.  2  bx -{- b'^ 

105  ^y  16  (^-^f^y+i^^-i^r 

15  bi/'  '               a*-b* 

12da%'^x'^  a- -a -20 


27  o'b\r*  a'  +  a  -  12 

a'-x'  —  16  u'^ 


18. 


a"-^^  iix^-{-dax-\-20a 

8       ''*''"  19           -^«  +  <^*    _ 

nx"-^  '  18tt-6a-^H-2tt» 

o{a^-U')  '         x?^xif 

10.       ^^-^  21.          26^-2^x 


2xy-\-2ij  'lbx^-\-^:h^x-^2b^ 

ax  -\-x:^  22     ^*  '*^  +  7  ajc  —  3  a;'** 

rtc*  4-  c=^  *   6a^  +  llaa;  +  3a:'^ 

aAr  -h_^  3  rf x<  4-  9  tf 6«  4-  6  M^ 

arx  -f-  r*  *        a^  .\^a%  —  2  d^b'^ 


24. 

25. 


6<tf  -I-  10  i<;  +  9  «c/  +  15  6rf 
(17^_|_9<.f/._l>,. -3(/ 


Remark. —  In  most  instances  hereafUT  it  will  be  desirable  to  use 
fractions  in  their  lowest  terms,  and  in  some  eases  it  is  necessary. 

II.      REDUCTION    OF    FRACTIONS    TO   MIXED    OR    ENTIRE 
QUANTITIES. 

161.  To  Reduce  an  Improper  Fraction  to  an  Entire  or  Mixed 
Quantity.     PriiK-iple  and  Kule. 

Since  a  fraction  is  an  expressed  division,  divide  the 
numerator  by  the  deuoniinator,  and  if  there  is  a  remainder 


162  TEXT-BOOK  OF  ALGEBRA. 

place  it  over  the  denominator.  This  amounts  to  the  same 
as  dividing  both  terms  by  the  denominator,  which  is  justi- 
fiable by  156. 

162.   Exercise  in  reducing  Fractions  to  Entire  or  Mixed  Quan- 
tities. 

1.    "^  +  ^^'       operation.       <^^  +  '^fia+x 


Operation. 

a-\-x  ^  ax-{-x 


X' 


rjf,^  ^  H        ; —  Ans. 

"^  a  -\-x 


25      147      75      1425  2  a'  -2ab-\-4:b^ 


"?        TTT' 


8'      19       13'      111  a-b 

ab  +  b^  .  22  a^b' -  S3  a'b' -\- 


a  11  a:'b 

X  —y 
5.    tJlf 

X 

4:X-  —  2x 


2x-  —  X  -{-1 


10. 
11. 
12. 
13. 


i  ax 


X 

a^ 

-1 

-hx'- 

-x"" 

1  ■ 

a  -\-x 
—  a  — 

ab  -f 

a% 

10 

ab-h 
a2  _  13  ax  - 

-Sx' 

„     a^x  —  3  ax^ 

7. : ;j—  14. 

w^  —  ax-  2a  —  3x 

^    x^-^Sx-]-2  jg    x'-^i/ 

x-\-3  '     x^y 

12  c^  4-  8  ac-x^  —  3  acx,  —  2  a/x^ 


3  c  +  2  ax~ 

163  Dissection  of  a  Fraction.  —  Principle  and  Rule.  — 
Divide  eacli  term  of  the  numerator  by  the  denominator 
(126,  2),  writing  the  quotients  in  the  fractional  form  con- 
nected by  their  proper  signs.  Reduce  each  fraction  to  its 
lowest  terms. 

164.   Exercise  in  Dissecting  Fractions. 
adn  -{-  ben  —  bdm  _  adn      ben       bdm 
bdn  bdn       bdn      bdn 

=    «   +  ^_  _  »     (159). 
bdn 


ALGEBRAIC   NOTATION.  lt)3 

2       6  <A- -.•)/;--(-  10  r- 
30  ubc 
abc  ~f-  bed  -|-  adc  -f  aftrf 
abed 

4    (/^^  -  ^0(^  +  y)  -  (^^^  +  ^^){p  -  y) 

(m  -  n){p  -  q) 
{a  -f  b){m  —  n)  —  {a  —  b)(m-\-  n) 

'  '~^,r^b'' 

165.  Fractions  written  as  Entire  Quantities. 

Hy  128,  2,  we  can  transfer  factors  from  the  denominator 
ot  a  fra(^tion  to  the  numerator  by  changing  the  signs  of 
their  exponents.  This  enables  us  to  write  any  expression 
in  the  integral  form. 

166.  Exercise. 

1.   — ^     By  changing  the  signs  of  the  exponents  of  a  and 
aif 

if,  we  have  a  ~  ^  bx^y  ~  ^ 

2  ^3-  6    ^  (^^  -  ^)^ 

>V2  '    4  (r/  -  bf 

3  tlL  6     ^l^^'*^ 


III.  -MIXED  QUANTITIES  TO  IMPROPER  FRACTIONS. 

167.  To  Reduce  a  Mixod  Quantity  to  the  Form  of  an  Im- 
proper Fraction.     rrin('ii>le  and  Kule. 

This  case  is  the  reverse  of  the  preceding.  There  the 
dividend  and  divisor  were  given  to  find  the  quotient.  Here 
the  quotient,  remainder,  and  divisor  are  given  to  find  the 
dividend,  which  being  found  is  written  over  the  divisor  for 
the  equivalent  friution.     Therefore,  the  entire  quantity  is 


164  TEXT-BOOK    OF   ALGEBRA. 

to  be  multiplied  by  the  denoiiiiiiator  and  the  numerator 
added  to  the  product  (or  subtracted  if  the  sign  before  the 
fraction  is  minus),  and  the  result  placed  over  the  denom- 
inator. 


168.   Exercise  in  reducing  a  Mixed  Quantity  to  the  Form  of 
a  Fraction. 

,     o       ,         ,2  ax  —  xif 
1.    3a-fy  + 1. 


Operation.     (3  a  +  y)  (x  —  y)  =  3  aic  —  3  ay  -\.xij  —  }f 
Adding  the  Numerator.         2  ax  —  xif 


i)  ((X  —  3  at/  —  y- 

Q       ,         ,    2aa?  —  xu       oax  —  oaa  —  ?/"'  i 

.'.  3  «  +  //  -|- ^  = -y iL.  Jus. 

x  —  y  x  —  y 


2.    2  +  3y/ 


y 


4y 


Operation.     (2  +  3y)  4y  -  (^  -  5)  ^  12yM^7,H:J  ^ 
4  y  4  // 

«.    The  minus  sign  before  a  fraction  changes  the  sign  of 
every  term  of  the  numerator  according  to  the  rule  for  sub- 
traction. (92,  1).     Thus,  -  —^  is  the  same  as  ~  f  "^  '^ . 
V     '    /  '4  2/  4:y 

3-   m  T^  4^. 

Remahk.  —  In  arithmetic  we  write,  e.g.,  4|  and  not  4  +  |,  while 
in  algebra,  «  +  ^  must  be  written  so,  and  not  a-^,  since  this  last 
would  mean  a  times  K  This  illustrates  a  difference  in  one  particu- 
lar between  the  arithmetical  and  algebraical  notations.  It  may  be 
said  that  when  no  sign  is  written  in  arithmetic  +  is  understood, 
while  in  algebra  the  sign  of  multiplication  is  understood  In  arith- 
metic the  numerator  is  always  added  to  the  product  of  the  denom- 
inator and  entire  quantity;  in  algebra  it  is  added  or  subtracted 
according  as  the  sign  before  the  fraction  is  +  or  — .     (Cf.  62,  b). 


AiJ.i:i'.l;Al(  NolAl'lON.  165 

4.  2a-'Jh-\-''  ~'^  10.    n-'-ax-\-x^ ~ 

3  a  -\-  X 

5.  (f-\-  11      <i- —  <i-\-h  Ir 

a  a  -{-b 

6.  o  -|-         12.    1.) r,  —  _ 

3x  Id 

7.  1+^^^  13.    i_<^'^-2«^+J! 

a  +  ^»  /^•■^  -f-  /y-^ 

8.  3  .r  -  i:fLzJL  14.    .r^  -f  .r^  -j.  .r  -f-  1  +  — -— 

oic  a:  —  1 

9.  (I  -\-0 — ^ —  15.    1  +  </ 5i /_  . 


IV. -REDUCTION  OF  FRACTIONS  OR  INTEGERS  TO  EQUI 
VALENT  FRACTIONS   HAVING  GIVEN   DENOMINATORS. 

169.  To  Reduce  a  Fraction  or  an  Integer  to  an  Equivalent 
Fraction  having  a  Given  Denominator.       Principle  and  Rule. 

Multiply  both  term.s  of  the  given  tnietion  by  the  quotient 
obtained  by  dividing  the  recjuired  dt  nominator  by  the  de- 
nominator of  the  fraction  (156;.     For  an  integer  the  denom 
inator  1  is  understood,  and  both  terms  are  multijdied  1  y 
the  recpiired  denominator. 

170.  Exercise  in  reducing  Fractions  to  Equivalent  Expres- 
sions having  Given  Denominators. 

1.  Reduce  a  to  the  denominator  d. 

()l)eration.     _  x  -  =  -— .     Arts. 
\       d        d 

2.  Reduce  —  to  the  denominator  .'*  7/^. 

n 

Operation.     3  ii^  -j-  ;*  =  3  m  ;  —  X  -—  =  — r-r  • 

3.  Keduce  5a%  to  the  denominator  ab'^d. 

4.  Reduce  |  to  the  denominator  56. 


166  TEXT-BOOK  OF  ALGEBRA. 

5.  Reduce  IL—  to  the  denominator  a-  —  h'^. 

6.  Reduce  5  {a  -{-  b)'^  to  the  denominator  (a  —  b)'-. 

7.  Reduce  ac  +  bd  +  ad  to  the  denominator  ab  -\-  cd. 

8.  Reduce  f-\-2x  to  the  denominator  a  -{-  b. 

9.  Reduce  — — — ^  '^  J  to  the  denominator  x^  -|-  ifi. 

x  +  ij 

10.    Reduce  — — — '^^—  to  the  denominator  a^  —  b'^  -\-  2  be 
a  —  b  -\-  c 

-g\ 

171.  To  Reduce  Two  or  More  Fractions  to  Equivalent  Frac- 
tions having  a  Common  Denominator,  (usually  the  lowest 
Common  Denominator).     Principle  and  Rule. 

Of  the  common  denominators  which  deserve  especial 
consideration,  there  are  two :  the  product  of  all  the  denom- 
inators of  the  fractions,  and  their  lowest  common  multiple. 
Taking  these  as  required  denominators  and  remembering 
the  process  of  169,  we  get  two  distinct  rules. 

1.  Find  the  1.  c.  m.  of  the  denominators  of  the  fractions, 
(called  the  lowest  common  denojninator) ,  and  multiply  both 
terms  of  each  fraction  (156)  by  the  quotient  of  the  1.  c.  d. 
divided  by  its  denominator.  The  results  are  the  equivalent 
values  of  the  respective  fractions,  having,  it  is  plain,  the 
1.  c.  m.  for  a  common  denominator. 

2.  Multiply  both  terms  of  each  fraction  by  the  product 
of  all  the  denominators  except  its  own.     (156.) 

Evidently  the  product  of  all  the  denominators  divided  by 
any  one  will  give  the  product  of  all  except  that  one. 

a.  The  1.  c.  d.  and  each  denominator  should  be  written  as  the 
product  of  their  factors.  Then  the  quotient  of  the  1.  c.  d.,  divided 
by  any  denominator,  is  the  product  of  all  the  factors  not  in  that 
denominator. 

In  addition  and  subtraction  of  fractions,  as  we  shall  see,  the  mul- 
tiplications have  to  be  performed  in  full  in  obtaining  the  numerators; 


ALGEHKAiC     >v^iAii"N.  107 

but  not  so  the  denominators.  Let  the  student  follow  this  ru'e,  never 
to  imtUqdy  fartor.s  toi/et/nr  unlil  it  is  seen  to  ftc  )U'C('!<s(trj/,  and  a 
great  saving  in  tiint'  ami  labor  will  thus  hv  niailc 

172.    Exercise  in  the  Reduction  of  Fractions  to  Equivalent 
Fractions  having  a  Common  Denominator. 


1. 


l_x'    1-x^    l-ar» 


( )peration.  —  Writing  the  denominators  in  the  factored 
form,  we  have, 

f_"x'    (l_a:)(l+x)'    (l-x)(i-\-x-{-x^y 

and  by  inspection  we  see  that  (1  -\-  x)(l  —  x)(l  -\-  x  -\-  x^)  is 
the  1.  c.  d. 

Now.  (Mi^Kki^KMi^J:^)^  (1  +  ,)(i  +  ,.), 

(l  -x) 

and  multiplying  the  numerator  by  this  quotient,  and  placing 
the  product  over  the  1.  c.  d.,  we  have, 

(^■^^yO--^^+^^         for   the    equivalent    value    of 

L±^,  the  first  fraction.  (169) 

In  like  manner, 
1  4-  ^^  _       (1  -f  a;^(l  ^x+x")       .  l4-ar>_ 
1  -  x'      (l-|-a-)(l  -  x){l  -{-x+x')'  1  -  x' 

(l-^x')(l-\-x) 


(l-\-xXl-x)(l-^x-\-x^) 


2     <•      ".>       1-      7  ^     a     c      e 

\)     12'    20'    10  '    V    d'    f 

6.    -^,     _!_,    -^ 
5  be     10  ac     ti  ah 

.     ^^      ^^  -     3  a;     2x       ?n 

2  a     oa^      n 


a 

b       1 

X 

x^'    ar» 

3a 

56 

lb 

'    21  r 

168  TEXT-BOOR   OF   ALGKBllA. 

^     9a     7b_     lla     7^i -^  b) 


Hx     36ic'     28  '         Ax 

9 

a  —  b     a  —  €     b  —  <i 

11. 

a-\-b 

a-b 

ab           ac           be 

a-y 

a^b 

10 

ab     cd      ef       <jli 
cd     ab      gh       ef 

12. 

1 

3 

4x-\-4:' 

13 

a^              1              a 

a}  —  x^'    a  —  X     a  -\-  X 

14 

X             y             X 

y 
-y 

x-^y     x-^y     X  ~  y     X 

1^ 

3              r> 

'lx^  —  2xy^'    ()(ir  +  //)"-^ 

16. 

1                   1 

1 

(^a  +  by     a  (a  -  by     b  (a'  -  P) 


SECTION   III. 
The  Fundamental  Opehations  in  Fkactions. 

173.  Addition,  Subtraction,  Multiplication,  and  Division  of 
Fractions.  The  Rules  for  these  operations  are  the  same  as 
in  Arithmetic. 

I. -ADDITION. 

174.  Addition  of  Fractions.     Principle  and  Rule. 

Let  —  =  a-,  and  -  =  y,he  two  fractions  having  a  common 
e  c 

denominator. 

a  —  xc  ^ 
Then,  ^  _  ^  ^  -  (Def.  of  division,  43) 

Adding  these  equal  quantities  (Ax.  1,  207) 

a  +  b  =  (x  +  y)e  (97) 

.-.       '      —x-\-y  (Def.  of  division,  43) 


ALGKlilJAlC    NofArioX.  109 

Thus,  when  two  fractions  have  a  conniion  denominator, 
the  numerator  of  the  sum  is  tlie  sum  of  their  numerators. 
This  demonstration  may  be  extended  to  include  any  number 
(»f  fractions  united  by  lx)th  -f-  and  —  si<;ns. 

RuLK.  Reduce  the  fnictions  to  equivalent  fractions  hav- 
ing a  lowest  common  denominator,  add  their  numerators 
and  i)lace  the  sum  over  the  conimon  denominator. 

175.    £zercise  in  Adding  Fractions. 

1.  —^  +  — ^     +;i 

X  —  1         X  -f-  1 
Operation. — We  write    1    for   the   denominator   of   the 
integer.     1  (x  —  1)  (or  -f- 1)  is  the  1.  c.  d.     The  equivalent 
fractions  are, 

xjx-^V) X  (x  -  1)  3(a;-l)(a;-f  1) 

{x^X){x-\y    (x  +  l)(x-l)"^  (x  +  l)(a:-l) 

Expanding  the  numerators  (171,  ^/)  and  adding  them,  we 
have, 

x^  -\- X  -\-  x^  —  X  ->^  ?t  x^  —  .'>  =  5  .r-  —  3, 

which,  placed  over  the  1.  c.  d.,  gives, 
5x^-3 


x-^-1 


Ans. 


4. 


2  ^4-^  4-11  7    ''^^-L^.l 

3  "  ~  "^    I    ?.?_ZlA_  8      ^^  ~  ^^  -I-  ^  "^-  4-  ^  ~  ^ 

5       '         2  (fO             ac            be 

8    ^12^4  ^~2~^:> 

ban  i          \) 

♦  6.   x-f- 1 —  11.    Add3x,xH ,  4x 

2                3  4                o 

Suggestion.  —  A<1<1  th«'  entire  parts  and  the  fractions  separately. 


170  TEXT-BOOK   OF   ALGEBHA. 


12.    4:a^  -\ 1 j \-  oa^  -{- 


13. 


3        '       3  '        7 

1        .        i  .^        n       ,   l-2n 


1 = —  16.    — ~ h 

ic  +  2        ic  +  3  n  —  ln'^  —  l 

14.        '''      +      i>  17.    1-^'  _p  1+^' 
?/t  4-i^      ^^  — p  14-^"^      1  —  x^ 

15     'i]ia^^J2^    ,   'nia  —  X  1 ,  1 


in  -j-  71         m  -\-  71  X  (x  —  y)       y  (^  -\-  ])) 

19.  Add^L-,  _L_,  ^_ 

a  —  h     h  —  c     c  —  d 

20.  Add       « 


1  _  a  '  (1  -  «)2 '  (1  -  a)3 
21.    -^^i-^+  ' 


ic"^  —  7  X  +  12       ic'^  —  5  ic  +  6 

22 ^  „™     4- ^ 

5  _^  ic  _  18ic' ^  2  +  5ic  +  2a;2 

1  2 


23.    Add 


a;-2_7a;  +  12'  cc2_4x  +  3'  £C'^-5ic  +  4 


24.    — -V  +  — V-  +  7-;V^,  +       "^ 


8-8ic       8  +  8ic       4+4  x'^      4-4ic* 

II.  -  SUBTRACTION. 

176.    Subtraction  of  Fractions.    Eule. 

Reduce  the  fractions  to  equivalent  fractions  having  a 
lowest  common  denominator,  and  subtract  the  numerator  of 
the  subtrahend  from  that  of  the  minuend  for  that  of  the 
difference.  If  several  fractions  are  connected  by  +  and  — 
signs,  reduce  them  to  the  lowest  common  denominator,  and 
add  the  numerators,  those  whose  fractions  are  preceded  by 
a  minus  sign  being  taken  negatively,  i.e.,  with  their  signs 
changed. 
■  See  })riii('i|)le  ex})lained  in  addition. 


ALGEBRAIC   NOTATION.  171 


177.    Exercise  in  Subtracting  Fractions 

1     ^-_  ^* 
L'd~r7 

2.    --1       -  +  - 


3.     1  _ 


3  o 

X  —  a 


4 

a-f  z; 

c-d 

a  —  b 

c-{.d 

1 

1 

b. 

_1 

a:  —  a 

^  x-Jfa 

6. 

l-\-x^ 

4x^ 

x-\-a  l-x^      1-x' 


,.  =.,l£^«_(  .,  +  --?) 


8.    2a-3x  + 


9.    X-     ="' 


a  —■  w       (  ^       .   X  —  <i\ 


a;  —  1      x-\-\- 


11. 


y^  —  y        y  —  1 

or*^  H-  aa;  -\- a}       x^  —  ax  -f  « /^ 


a"  —  a'  x"  -|-  f/ 

a  —  2  x  rr''  —  8  ar"^ 

4  5 


13. 


14. 


4_7a__2a2       4  —  5a-6a^ 
2.2  1 


15.    1  + 
16. 


y^  _  3a.  4.  2   '  x^-x  -1       X-  -  1 


4ab 

,2 


(a  _  ft)  (6  _  r)    ^  (^  -  r)  (A  -  a) 

SuooESTiON. — We  saw  in  142,  14,  that  a  —  b  and  h  —  a  are  the 
same  quantity,  but  with  opixjsite  8ign.s.  If  we  change  the  ft  —  a  in 
the  s<'conil  denominator  (to  make  it  the  same  as  ri  —  b  in  tlie  first 
denominator)  the  sign  of  the  whole  denominator  is  changed.  For, 
changing  the  sign  of  one  factor  of  a  product  changes  the  sign  of  the 


172  TEXT-BOOK   OF   ALGEBRA. 

product  (107,  1).     Now,  to  leave  the  fraction  as  it  was  before,  we 
change  the  sign  of  the  fraction  (157).     Thus,  we  get, 

a2  />2  a2-62  a-\-b 


(a-bXb-c)      ib-c)(a-b)      (a-6)(6-c)      b-c 


(160) 


17. 


b  —  a       a  —  2b 3cr  {a  —  b) 

X  —  b        b  -\-  X  b'^  —  x'^ 

l-2«^l+2;r  ^  4rx^-l 

19.  ^ -  + i! ,+ 


(^a-b)(n  —  e)        (b  -  r)  (b  -  a)       (e,  -  a)  {e  -  b) 

Suggestion.  —  For  the  sake  of  system,  it  is  convenient  to  have 
regard  to  the  cyclic  order  of  the  letters.  To  do  this  we  think  of 
them  as  placed  in  a  circle  so  that  they  follow  one  another  in  regular 
order,  the  last  being  succeeded  by  the  first.  Thus,  if  there  are  four 
letters  a,  b,  c,  cZ,  we  go  from  a  to  b,  from  b  to  c,  from  c  to  d,  then 
from  d  to  a  again.  Of  the  six  expressions  in  the  example  only  three 
are  written  in  cyclic  order,  viz.,  a  —  b,  b  —  c,  and  c  —  a.  Changing 
the  others  to  this  order  as  in  the  suggestion,  Ex.  16,  we  get, 

a2  ?>2  c2 


(a-b)(c-a)      (b-cXa-b)      (c-a)(b  —  c) 
the  1.  c.  d.  of  which  is  now  easily  seen. 

20.  r-jj-q        ^       q  -^r-p       ^      r -\- p  -  q 


{P  -  q)  (r  -p)       (q  -  r)  (q  -  p)       (r  -  p)  (r  -  q) 

21.  yH-'^  _  ^  +  ^  _^ x^y 

(^x-y){z-x)       {y-z){y-x)       (x  -  z)  (z  -  ij) 

Suggestion.  — In  the  third  expression  both  factors  of  the  denom- 
inator will  have  to  be  changed,  which  will  not  change  its  sign 
(107,  1). 

22.    - -f 1 4-  ^ 


(b  —  g){c  —  a)       (b  —  a){a —c)       (a  —  b)(b  —  c) 

23.    i ^  1  _    1 

X  (x  -  y) {x  -  z)       y  (y  -  x)(y  -  z)       xyz 


AlJJKIUiAlC    NOTATION.  173 


III.  -  MULTIPLICATION. 

178.  Multiplication  of  Fractions.    Rule.    (Ex.  21,  Art.  255.) 
Factor  the  terms  of  the  fractions,  cancel  common  factors, 

and  then  multiply  the  numerators  together  for  the  numera- 
tor of  the  product  and  the  denominators  for  the  denomi- 
nator of  the  product. 

a.  Integral  factors  are  to  be  regarded  as  numerators  over  tin* 
denominator  1  undei-stood. 

h.  Quantities  to  be  multiplied  (or  divided)  must  lirst  be  reduced 
to  either  integral  <}uantities  or  simi)le  fractions. 

179.  Exercise  in  the  Multiplication  of  Fractions. 

1.  -_L'  X  ()  a'     Operation.      _  X  ^^—  =  4  ah. 
«5  a 

2.  ,.  X  -^  X  •'/ 


3.  -'^'X"* 
cd 

4.  4j-yX 

4  jrt/ 

8.    !"l^J':  X  (nr'  4-  2  mn  -h  //-) 
ft/  -\-  n 


3^.        1 

5. 

loa'V 

24  rV 

6. 

-<') 

7. 

4.1a-7/V'' 

24;^r//V 
180  a'^6c« 

27a:y;. 

9. 

17.^^-4 
*2abx-\-b 

Xab 

LO. 

x*-a 

-X3y 

LI. 

•t'  +  -x 

5x 

3  2a- -1-1 

d(x-\-f/)(x-ij)         ^    ^^^ 


13. 


\^  a  J       X 


174  TEXT-BOOK   OF   ALGEBRA. 


5  a     ^  2  -  a 

I/  — 2 

-„     a -\- X      a  —  x  <i'^ -{- x^ 

16.    X X  o 


3  {a'  -  x^) 

,„    a" -121       a-\-2 
17. V 

x^-\-o  x  -\-6  x^  —  2  a.'  —  3 

18. '.T-  .  -     X  - 


x'  -  9 


19 X 


2  ax  —  2  XT 


^^    ic^- 13  a.' 4-42      x2-9:i'-f20 
20. ^ — J- — X 2 — ^r- — 

a?-  —  oic  x^  —  bx 

X  -\-l  X  —  1 

21.  —  3  (t  X ^7^ X r  7 

22.  ^'  +  1       ^  +  -\.    ^-1 

23.  i^.  +  ^>^0^'+^0' 


25.  (l_.,  +  .,-2)^        1_^1 

y      ^   a    '    a^ 

26.  Multiply  together 

ic^  —  X  —  20      ic^ — X  —  2      ic^  +  oa? 

27.  Multiply  1  +  1  +  -Uy  1  _  1  +  -1 

a       0       G         a       0        G 


ALGEBRAIC   NOTATION.  175 


IV.  -  DIVISION. 


180.  Division  of    Fractions.     Kule.  —  Invert   the    divisor 
ml  proceed  as  in  niultiplit'ution.     See  Ex.  22,  Art.  255. 

181.  Exercise  in  Division  of  Fractions. 


1. 

Divide  ^'''  by  ^'  "*.    Inverting  tl 
9m         3 

2a 
3 

2. 

,       3d 

—  ah  -^      -- 

2  (lb 

3. 

4. 

''^  3b 

6. 

'      ^oh''            27  ^2  n        ahe 
40c    ^       81rfJ   •  14rZ« 

6. 

_8w  *    21n^xhj  \  '^  81m7i 

7. 
8. 

[b^><d'<Ly  b^ 

a  4-1      «2_i 

(f        '        a 

9. 

-4-  (//  -  ^) 

^  -\- !f 

10. 

1                  1 
a«  _  i«    •  a  _  6 

11. 

Divide  f  ~^by  a+c 

0  -f"  ^ 

12.    Divide  1  +  x  by  i  (1  -f-  .r* 


(178,  a) 


176  TEXT-BOOK   OF   ALGEBllA. 

14.    6  (a  -^x)  +  2  _  4 

3  *    3  (a  —  if) 

1,^     f;^-       ^       ,   V2a'~]      r        .2^2"] 

16.     '^^  ''•^'    ^    •    '''  ^'^'  ""  '^^ 


\//      -^v      \.r     //     ^/ 

-('•-:-)-('-:o 

U-^  +  24x  +  128 '^         .T^-64      J*    ic*'  +  4a'  +  16 
20.    r^<^^  —  3  a'b  H-  3  6?//'^  —  b^  _^  2  ^^/>  —  2  ^'^l    ^   aM;^^> 

[_  ^2  _  ^-2     —   •  -      3      J  ^  7t^ry 

182.  Complex  Fractions.  —  When  the  division  of  one  frac- 
tion by  another  is  indicated  in  the  fractional  form  by 
writing  the  dividend  over  the  divisor,  there  resnlts  what 
is  called  a  complex  fraction.     (155,  1.) 

To  simplify  such  fractions  write  the  numerator  as  the 
dividend,  and  the  denominator  as  the  divisor  in  a  problem 
in  division  of  fractions,  and  proceed  according  to  the  rule. 
(180.)  Or,  what  comes  to  the  same  thing,  multiply  the 
extremes  for  the  numerator  and  the  means  for  the  denom- 
inator, and  then  simplify  the  resulting  fraction. 

183.  Exercise  in  the  Reduction  of  Complex  Fractions  to 
Simple  Fractions. 


1.     7- 


1/ 


a 


5  —  c       7  —  y       5  —  c  a  5  a  —  ac 

1  (»  _^  tL  __  ^  s/  =  

'  *'  X        '        a  X  1  —  ij        1 X  —  xy 


ALGKin:Aic  NcnwriMX.  177 


a  -f  h  »>  iU 

//  7  axy 

^<r  -j-  f*  3  a* 

hx  —  <t  *      3  a 


1 


SoiA'Tiox.  —  We   must  rediu'e  both  the  iiuinerator  and 
the  denoiuipator  to  the  form  of  simple  fractious  hy  167. 


1 


z 

_. ^ (l^v  re(hu3iu«;  the  denominator.  167.) 

xj/z-^x-^z 

y«  +  i 

Now  y  ^  «y  -f-  1  ^   a;//g  -\-x-^z  ^ 

(^y  -h  1)  (j/z  +  1) 

Remark.  —  Reduce  continued  fractions  such  as  the  denominator 
above,  step  by  step,  commencing  with  the  last  denominator  (167), 
until  simple  fractions  are  attained  in  both  terms. 


1T8  TEXT-BOOK  OF  ALGEBRA. 

6 


X 


.T  —  1  + 


^  a;  -  2  + t: 

'  a;  —  6 

1 


8. 
9 

n 

1 

1     -  A 

10. 

1  + 

1 

^]  -  ^  6' 

•+ 

2) 

11. 

n  +  i  (.^ 

,-  — 

^>; 

14.     9 


Uo^  + 


X 


4 


^ 2x-\ 

15.     ^  +  1  ^       ^_1 

^   +2       2 

1  1_ 

16     1  +  ^^-        1  —  <^ 
1  1 


+ 


x^ 


1  —  a    ^  1  -\-  a 

i  +  'i^    i  +  "^-l 

^        a  —  x       -j        a^  —  x^ 

a  -\-  X  a'^  +  x^ 

2x  3 


17. 


x-'d 

12.  TT-r  18.  1  + 


1  + 


x-?y  "^1_^ 


184.   General  Exercise  in  Fractions.  —  Siiiii)lify  the   quan- 
tities of  tliis  article. 

^  ax  27  a  -\-  a^ 


(fx-  —  ax 


18  a  -  6  a^  -{- 2  a^ 


5  a^l,  +  10  a%^  ^      .7-2  _  5  x-\-6 


:Ui'b-^ -\- 6  ab^  x^-1  x+12 

^     2x  —  ?yy,x^2ii      :^  .T  —  2  ?/ 

5. -\- — "^   —  _ '- 


alokbi:ai('  Nor a'iion. 


179 


6. 

7. 
10. 
11. 
12. 
13. 
14. 
15. 
16. 
17. 
18. 
19.    {  1 


r>  + 


a  *  1 

</  —  1  a{(i  —  1) 

1  .  1 


4  «a^  —  5  x^ 
a  —  X 
a»  -  3  M  -\-  3  ab^  —  b* 


8.    a  -\-  X  — 
9. 


^2  -  2  </^  4-  b' 


+ 


(a  -  b)  (b  — c)      (a  -*)  (a  -  c) 
3a;-l   ,  2a;+5^  4a;-l 


x  +  2     '  2.r-f-4  '  6«  +  12 
1 


^       4-  _  _^.__. 
./■  -f-  o  //      x^  —  9  //- 

ar^~2a;  +  l 

4a'' -1       *  2^+1 
25rt'^--&^    X  ^(  — "t^ 


20. 


;>,^-2  + 

Sx-l 


21. 
22. 


/>-7  _j.  ^--/^  4. 7_ir 


y^Y 


y>;- 


qr 


a^  —  ar*  ^  (a  -f  xy 
a^  -\-  xr^       {a  —  x)' 


180  TEXT-BOOK  OF  ALGEBKA. 


23.    3V_5y/-  +  7_(>.r-^-ll 
xy^  xy^  x^y 


24.    ^  +  ±_ 


25.    ^^' 


f^lytlffl 


2g     4a^-16^>^^  5  a 


a -2^  20<^-^  +  8a^  +  8^2 


27.    ^xi-^ 


rtQ  ma^  —  7ix 

<60. 


(a  -j-  ^)  (m  —  w) 


.x-8  +  1 

1-x^         1  -y^  (  X      \ 

'  ~  1  +  7/  ^  x^  +  x^  -y^  +  i-xj 

32 ^ 

'    3  (1  _  ^)  "^  3  (^  _  1)  "^  i 
34.       _^      ..+  ^ 


7i 


x(x^  —  y^)      y{x^-\-y^) 


35.  X 

3  %  c^  —  x'^  d^  —  ax 


X  -\-  a        x-\-3  a       a  —  x       x  —  3a 
2-bx  _3H-a;,2a;(2a;  —  11) 


ALGEBUAIC   NOTATION.  181 


38.  l+«  ^  1+*  _L  1-f^ 


(,,  _  A)  (a  _  r)        {b  —  c){b-a)        {c  —  a)  {r  —  b) 


ar-hl       (^.4-l)(x  +  2)       (a;  +  l)(a;-f2)(x  +  3) 

*   ^-^f      ^'^y'       y'-x'       {x  +  y)ix^^y^) 
ni^  -\-  n^ 

n  rti^  —  n^ 

n        m  A 

42.  3a-[/>+S2«-(^-.)n  +  |+     2-eTT- 
1 

43.  a-  4-  ^ 


■+3-^; 


'-■1+47     '-^tf 

a;  -f  1  x^  —  1 


X 


x--^^--        ^{x-l){x-2) 


45. 

If 


46 


x-  1 


47.  f-+l4-lU(-4.iL  +  ^) 
\x       y       z  I      \x        y       z  J 

48.  p  +^  4-  ^  ~^1  -^  r*  J-  ^  _ ''  ~  ^1 

49    1    r  ^    I  ^+M 

•«(T+~)(i-4)*(:--)(7-l) 


SECOND   GENERAL  SUBJECT.^  —  SIMPLE 
EQUATIONS. 


CHAPTER   XIV. 

SIMPLE     EQUATIONS     CONTAINING    ONE    UNKNOWN 
QUANTITY. 

185.  Definition.  —  An  Equation,  is  an  Expression  of  the 
Equality  of  two  Quantities. 

By  this  is  meant  that  the  numerical  values  of  the  two 
quantities  will  be  found  upon  reduction  to  be  the  same 
number,  integral  or  fractional.  (See  55,  where  the  mean- 
ing and  use  of  the  equality  sign,  =,  was  explained.) 

Equations  considered  in  a  general  way  may  be  regarded 
as  expressing  the  relations  of  numbers. 

SECTION  I. 
General  Definitions. 

186.  Equations  are  of  two  kinds,  Identical  and  Conditional. 

187.  An  equation  in  which  one  side  is  but  the  develop- 
ment of  the  operations  indicated  in  the  other,  or  is  exactly 
the  same  on  both  sides,  is  called  an  Identical  Equation. 

Thus,  (x  +  3)  (x-9)  =x^-6x-  27, 
and  ax  -\-  b  =  ax  -\-  b, 

illustrate  the  two  kinds  of  identical  equations. 

1  A  discussion  of  powers  and  roots  as  forming  tlie  remaining  subjects  in  the 
"Algebraic  Notation"  ought  theoretically  to  be  given  before  taking  up  equa- 
tions.   Other  reasons  exist,  however,  for  changing  this  order. 

182 


siMi'i.i:  i:(,n  AiioNs.  183 

The  examples  given  in  addition,  subtraction,  multiplica- 
tion, division,  factoring,  and  t'rac;tions  set  equal  to  their 
answers  would  be  equations  of  this  sort. 

a.  A  prime  characteristic  of  identical  equations  is  the  property 
that  any  letter  may  have  any  value,  provided  it  has  the  same  value 
on  both  sides  of  the  equation.  This  is  true  because  the  two  sides 
of  the  equation  are  either  actually  or  virtually  the  same  quantity, 
and  conse<iuently  if  for  a  letter  the  same  number  be  substituted  on 
both  sides  the  two  results  are  the  same  (84). 

h.  To  indicate  the  relation  of  identity,  or  to  show  that  one 
expression  stands  for  another  (113),  the  sign  "=5"  (read  idrntiralh/ 
equal  to)  is  used.  Whenever  the  distinction  remarked  in  this  article 
needs  to  be  brought  to  mind  the  identity  sign  will  be  used.  All 
identities  are,  of  course,  equations,  but  not  all  equations  are  iden- 
tities. 

188.  An  equation  in  wjiich  a  definite  number  onlij  of 
numerical  values  of  a  letter  will  imswcr  to  maintain  the 
equality  is  called  a  Conditional  Equation. 

This  is  the  8i)ecies  of  equation  about  to  be  investigated. 
For  examples  in  which  the  unknown  letter  has  but  <nie 
numerical  value,  see  47  and  85. 

189.  An  Equation  of  Condition  is  regarded  as  containing 
along  with  known  quantities  an  unknown  quantity,  (repre- 
sented by  a  letter)  whose  numerical  value  is  to  be  found 
such  that  when  it  is  substituted,  the  equation  may  be  veri- 
fied.    (55,  a.) 

Thus,  if  X  represents  some  unknown  quantity, and  i\  -{-  ".»./• 
=  51,  then  9  a;  =  51  —  G  =  45,  and  x  =  5. 

Therefore,  substituting  the  value  of  a-,  (J  -f  9  X  5  =  51, 
which  is  a  true  ecjuation.  If  any  value  other  than  5,  as  7 
or  12,  etc.,  is  substituted  for  sc,  the  result  is  not  a  true 
equation. 

a.  Substituting  the  value  of  the  unknown  (piantity  for 
it,  and  then  performing  the  operations  indicated,  showing 
the  two  sides  of  the  equation  to  be  equal,  is  called  verifying 


184  TEXT-BOOK   OF   AI.GKlUlA. 

the  equation.     When  an  equation  is  thus  tested,  and  the 
sides  found  equal,  the  equation  is  said  to  be  satisfied. 

b.  The  simplest  form  of  the  equation  occurs  when  the 
unknown  number  stands  alone  on  one  side  of  the  equation. 
Such  is  the  usual  form  of  the  equation  in  arithmetic,  where, 
after  the  equality  sign,  an  interrogation  point  is  often  sub- 
stituted for  the  answer. 

Thus,  16  +  9  X  3  -  12  =  ? 

In  this  case  all  that  needs  to  be  done  is  to  find  in  the 
simplest  form  the  numerical  value  of  the  given  side  of  the 
ec^uation.  In  algebra  the  unknown  quantity  may  appear 
in  any  part  of  either  side  of  an  equation,  or  on  botli  sides 
at  once,  and  to  find  its  value  in  such  cases  is  far  more 
difficult  to  accom[)lish. 

c.  The  Equation  takes  such  an  important  place  in  algebra 
that  many  writers  define  algebra  itself  as  the  science  of  the 
equation.  The  Germans  call  our  arithmetic,  Recltnung, 
reckoning,  and  literal  arithmetic,  Buchstahenrechmmg,  i.e., 
reckoning  with  letters,  reserving  the  title,  algebra,  for  the 
study  of  equations.  The  term  Arithmetik  is  also  used  by 
them  for  literal  arithmetic. 

190.  The  Right  Side  of  an  equation  is  also  called  its  Right 
Member,  and  the  Left  Side  its  Left  Member. 

'    191.    A  Numerical  Equation  is  one  in  which  the  known 
quantities  are  represented  by  figures. 

192.  A  Literal  Equation  is  one  in  which  the  quantities 
regarded  as  known  are  represented  by  letters,  or  by  letters 
with  figures. 

193.  To  Solve  an  Equation  (i.e.  to  loose  the  unknown 
quantity  from  the  others)  is  to  find  the  value  of  the  un- 
known quantity.  This  value  is  called  the  Root  of  the 
equation. 


SIMIMJ;    i:<,M   A  llONS.  l»o 

194.  There  are,  as  has  been  stated,  two  kinds  <>t  (juaii- 
titirs  jn-eseiit  in  a  conditional  ecjuation,  the  known  and  the 
unknown.  Tiie  latter  are  commonly  represented  by  the  last 
letters  of  the  alphabet,  (x,  y,  or  z),  while  the  former  are 
represented  by  figures,  and  by  the  first  letters  of  the  al- 
phabet.    This,  of  w)urse,  is  purely  conventional. 

195.  The  Degree  of  an  Equation  is  the  same  as  the  highest 
|)o\ver  of  the  unknown  (juantity;  or,  in  equations  contain- 
ing more  than  one  unknown,  it  is  the  same  as  the  degree  of 
the  term  (see  77)  containing  the  greatest  exponent,  or  the 
greatest  sum  of  exponents  of  the  unknown  quantities. 

Thus,  ox  +  6  =  91  is  of  the  first  degree. 
ax'  -\-hx  =  c   is  of  the  second  degree. 
11 X  -\- {)  j-^  -\- 7  y-  =  4li   is  of  the  third  degree. 
^z"*  -{-3z'"-^  —  2z '» -  -  =  7    is  of  the  m""  degree. 
G  x»  +  9  xY  +  14  //*  =  21  J-//   is  of  the  fifth  degree. 

196.  Equations  of  the  First  Degree  are  also  called  Simple 
Equations.     Sec  licading  of  this  cha})tcr. 

".  Of  all  equations  those  of  the  first  degree  ;irc  thr 
easiest  to  solve.  The  treatnient  of  sinijde  equations  only, 
and  not  of  equations  in  general,  will  l)e  taken  uj>  at  this 
time. 

SECTION   II. 
Aloebrak'.vi.  Mktuoi)  of  Thk.\tmk\t. 

197.  Nature  of  the  Treatment.  Tlie  metliod  by  wliich 
the  problems  of  47  and  85  were  solved  was  of  a  special 
nature,  similar  to  what  is  called  analysis  in  arithmetic. 
Scarcely  any  two  were  reasoned  out  in  the  same  way.  The 
treatment  to  which  the  equaticm  is  now  to  be  subjected  is 
of  a  very  different  character,  more  comprehensive  and  more 
scientific. 


186  TEXT-BOOK   OF    ALGEBRA. 

I. -LOGICAL   TERMS. 

198.  A  Proposition  is  a  statement  presented  for  considera- 
tion. A  proposition  may  be  true  or  false,  and  may  therefore 
be  proved  or  refuted. 

199.  Propositions  are  of  two  kinds,  theorems  and  problems. 

200.  A  Problem  in  algebra  is  a  question  proposed  for 
solution. 

201.  A  Theorem  is  a  proposition  to  be  proved  by  a 
demonstration. 

202.  A  Demonstration  is  a  chain  of  reasoning  by  which 
the  truth  or  falsity  of  a  proposition  is  made  evident. 

203.  A  Corollary  (cor.)  is  a  truth  inferred  from  a  prop- 
ositicjn  or  from  something  in  its  demonstration. 

204.  A  Scholium  (sch.)  is  a  remark  concerning  a  proposi- 
tion or  its  proof. 

205.  An  Hypothesis  (hyp.)  is  a  supposition  made  at  the 
beginning  or  in  the  course  of  a  proof  or  solution. 

206.  An  Axiom  (ax.)  is  a  self-evident  truth. 

II. -AXIOMS. 

207.  The  following  are  the  Axioms  used  in  the  Solution  of 
Equations. 

<(.    Addition  and  Subtraction. 

1.  If  the  same  quantity  or  equal  quantities  be  added  to 
equal  quantities,  the  sums  will  be  equal. 

2.  If  the   same    quantity  or  equal  quantities  be   sub- 
tracted from  equal  quantities,  the  remainders  will  be  equal. 

b.    Multiplication  and  Division. 

3.  If  equal  quantities  be  multiplied  by  the  same  quan- 
tity or  equal  quantities,  the  products  will  be  equal. 

4.  If  equal  quantities  be  divided  by  the  same  quan- 
tity or  equal  quantities,  the  quotients  will  be  equal. 


SIMPLE    lOQl'ATIONS.  187 

c.    Powers  and  Roots. 

5.  If  e(iual  (luaiitities,  or  the  same  quantity  be  raised 
to  the  same  power  the  products  are  equal.  * 

6.  If  equal  quantities,  or  the  same  quantity  have  tlie 
same  root  extracted,  the  results  (there  being  more  than  one; 
are  resjiectively  equal. 

7.  Quantities  which  are  equal  to  the  same  quantity  are 
equal  to  each  other. 

8.  General  Axiom.  —  If  the  same  operation  be  per- 
formed on  two  e(pial  quantities,  the  results  will  be  equal. 

XoTK.  —  It  will  often  be  convenient  to  refer  to  the  aildition  and 
subtraction  axioms  tiKjcther,  as  axiom  «,  and  to  the  multiplication 
and  division  axioms  as  axiom  h.  When  the  addition  axiom  alone  is 
referred  to,  it  will  be  by  the  number,  1,  and  so  for  the  others. 

III.  -  SOLUTION  OP  EQUATIONS. 

208.  Solution  of  Equations  of  the  Form  ((jc  =  b.  To  solve 
this  equation  x  must  In*  made  to  stand  alone  on  one  side  of 
the  equation.     (193.) 

We  have 

ax=:h  (Hypothesis,  205) 

Dividing  both  sides  of  the  equation,  that  is  the  two 
equals  by  a, 

a;  =  -  (Axiom  4) 

a 

Notice  that  this  solution  is  genera,  in  its  nature.  Kor,  a 
may  stand  for  any  co-efficient  of  x,  and  h  for  any  quantity 
whatever  on  the  right  side  of  the  equation  (113).  This 
element  is  characteristic  of  all  algebraic  solutions,  and  must 
l)e  remembered  to  understand  them.  Notice  further  the 
use  of  the  axiom  to  accom})lish  the  end  desired. 


188  TEXT-BOOK   OF   ALGEBRA. 

209.  Exercise. 

1.  9.r=36. 

Dividing  the  two  equal  quantities  by  9,  by  axiom  4,  the 
quotients  are  equal.     .-.  ir  =  4,  answer. 

2.  H)x  =  64.  8  m  =  nx. 

3.  9:^=144.  9.  45?/ =  120. 

4.  \\x  =  29.  10.  17  a;^  =  3  «,  ^  =  ? 

5.  «^x  =  16.  11.  6  a-  +  5  X  =  33. 

6.  3r^/'  =  7c.  12.  7x  — 3a;  =  33. 

7.  35  =  5 J'.  13.    2ic  +  3.x  +  9^  =  28. 

14.  2j'-hl4.T-13:r  =  29-14  +  12  — 3.     (87,4.) 

15.  9  //  +  7  y  -  2  //  +  S  //  —  //  -  2  ^  =  36  +  10-8. 

16.  25  +  S- 3  +  20 -9+1  =-  +  2;v  +  llrv  —  7;s. 

17.  {a  +  />  +  <')  -r  =(l  +  ('  +/■ 

18.  m.r  +  //.'•  +  px  +  y./-  =  /•.      (97.) 

19.  (i.r  +  />■/'  =  <i-  —  (r. 

20.  S.r  +  3.r  — ir)./- =  28. 

XoTK. — It  often  happens  that  the  coefficient  of  the  unknown 
becomes  minus  on  the  left  side  of  the  equation.  When  this  is  the 
case  the  equation  shouhl  be  divicU^d  through  (Ax.  4)  by  the  coefficient 
of  X  taken  with  its  negative  sign.  Thus,  in  the  example  before  us, 
upon  reduction 

-  4  .r  =  28 

X  —  —1     (by  dividing  through  by  —  4.) 

21.  19.?/ -(37// -12//)  =  14  — 34-4. 

22.  11//  —  (5  //  —  3  y)  =  25  —  88. 

23.  —  //  =  (f  —  J). 

24.  {<i  —  h)  ./•  =  <i^  —  h^. 

210.  Solution  of  Integral  Equations  in  which  the  known 
and  unknown  quantities  a])|)ear  })r<)niiscuu)usly  on  botli 
sides  of  tlie  equation.     Ti-ans])()siti()n. 


SIMPLE  EQUATIONS.  189 

To  show  how  such  equations  can  be  solved,  we  may  take 
the  equation  ax  —  h  ^=  ex  —  d  in  which  both  a  term  contain- 
ing X  and  a  term  independent  of  x  appear  on  each  side. 

ax  —  b  =  ex  —  d.  (Hyp.) 

Adding  b  to  each  member  of  the  equation, 

ax  —  b^h  =  ex-\-b  —  d.  (Ax.  1,  207) 

Or,        ax  =  ex  -\-b-  d.  (87,  4) 

Again,  subtracting  ex  from  each  side  of  the  last  equation. 

ax  —  ex  —  ex  —  rx  -\-  b  —  d.  (Ax.*2) 

Or,        ax  —  ex  =  b-d  (87,  4j 

(a-'C)x  =  b  —  d  (97) 

x  =  ^^.  ,208) 

a  —  e 

The  solution  sliows  that  —  b  was  transferred  or  '•  trans- 
posed" from  the  left  member  to  the  riglit  and  its  sign 
changed  by  adding  -\-b  to  both  sides.  Similarly  ex  was 
transjK)sed  from  the  right  member  to  the  left  and  its  sign 
changed  by  subtracting  ex  from  lx)th  sides.  Evidently  this 
would  apply  equally  well  to  any  quantities  on  either  of  the 
sides  of  the  equation,  and  may  be  stated  as  a  theoiem  as 
follows :  — 

Theorem  op  Transposition.  —  Any  quantity  may  be 
transposed  from  one  side  of  an  equation  to  the  other  by 
changing  its  sign. 

Scholium.  —  By  this  theorem  all  the  terms  containing 
the  unknown  quantity  can  be  transposed  to  the  left  member, 
and  all  the  independent  terms  to  the  right  meml)er  of  an 
equation.  When  this  is  done,  by  uniting  the  terms  on  each 
side,  the  equation  may  be  solvfjd  by  208. 


190  TEXT-BOOK   OF   ALGEBRA. 

211.    Exercise. 

1.  i)x  =  3^  +  24. 

By  subtra(3ting  3a?  from  each  side  we  get 

3  ic  =  24,  whence  x  =  S.  -  (208) 

2.  9  a^  -  2  =  7  X  +  16. 

Model  Soll tion.     9x  —  2  =  70?  + 16 

9^—7.^  =  2  +  16  (Ax.  «) 

2^^  =  18  (86) 

ic  =  9.     Ans.  (Ax.  4) 

The  student  should  learn  to  follow  this  model,  placing 
the  appropriate  reasons  for  the  different  operations  in 
])ar(Mitheses  at  the  right  margin.  Positive  terms  are 
transposed  by  suhtydctlny  (Ax.  2),  and  negative  terms  by 

(tddiiKl  (Ax.  1). 

3.  8 :«  —  7  +  3  .X  =  15  +  2  £c  —  13. 

6.  ax  =  inx  —  n. 

6.  ()  X  +  4 ./:  —  13  —  2  ./•  -  3  =  0. 

7.  ax  -J-  hx  -\-  ex  —  r/  =  0. 

8.  iihx  -\-  a  -{- hx  =  h  —  px. 

9.  mx  —  nx  =  0. 

Solution,     (m  —  7i)x  =  0  ;  .-.  x  =  0.  (Ax.  4) 

10.  ir)(.-r-l)+4(^+3)  =2(7  +  it;). 

SuG(iKSTiON. — First  perform  tlie  multiplications  indicated. 

11.  118  -65^-123  =  15£c  + 35 -120a;. 

12.  7x  —  5[x  —  ]7  —  6(x—3)\']=Sx-i-l. 

(106  and  102) 

13.  (x  +  3)  (2x  +  3)  -  14  =  (x  +  1)  (2ic  +  1). 

Remark. — This  equation  has  the /on/?  of  one  of  the  , second 
degree  (195),  but  upon  multiplication  and  transposition,  the  ic^'s 
cancel  leaving  a  simple  equation. 


SI.MIM.K     1.<,M     \  lloNS.  191 

14.  3(a--l)'^-3(^^-l)=:r-  15. 

15.  2  (j-  -\-2)  (x  —4)  =  X (2x-\-l)-21. 

16.  2b  —  (0-\-c)x  =  (h^c)x 

-  {b  -\-  c)  X  -^  {b  —  v)  X  =  --2b  (Ax.  2) 

-2bx  =  -2b  (106,86) 

x  —  \.     Ans.  (Ax.  4) 

17.  a-\-2(1  ^^x  —  {2b-\-4:c)  =  a- —  (7  <•  -  ()</). 

18.  ii'-x  -f  hx  —  / •  =  b-x  -f  r.r  —  d.  (97) 

212.  Solution  of  Fractional  Equations  in  which  the  uu- 
kiiown  in;iy  appear  in  any  part  of  either  side  of  the  erjua- 
tion.  Such  equations  can  always  Ix*  reduced  to  the  integral 
iorni.  For,  an  equation  may  be  multiplied  through  by  any 
factor  whatever,  and  the  new  efjiiation  will  hold  true  (Ax.  3). 
Now,  if  the  equation  be  multiplied  through  by  a  common 
multiple  of  all  the  denominators  of  its  terms,  such  denomi- 
nators will  all  cancel  out  of  tlie  new  equation.  (See  defini- 
tion of  common  multiple  149,  a.) 

Theorem.  —  Any  equation  can  be  cleared  of  fruitions  hif 
iinilfiidi/infj  both  Its  members  by  a  common  multiple  of  its 
d  I' nominators. 

Corollary.  —  The  signs  of  all  the  terms  of  both  mem- 
bers of  an  equation  may  be  changed  from  +  to  —  and  — 
to  4-,  for  this  is  equivalent  to  multiplying  both  members  by 
—  1,  which  is  i)ermissible  by  Ax.  3. 

S<!H.  I.  —  Any  common  multiple  of  tlie  denominators  will 
answer  to  clear  an  equation  of  fractions.  It  is  greatly  jjref- 
erable,  however,  to  multii>ly  through  by  the  lowest  romino)i 
denominator,  thereby  obtaining  simpler  expressions,  and 
that  more  readily. 

ScH.  IF.  —  Tliis  proposition  takes  any  equation  ami  ex- 
plains how  to  clear  it  of  fractions.  It  is  then  ready  for 
tran8iX)sition,  and  ultimately  for  solutiou. 


192  TEXT-BOOK   OF    ALGEBRA. 

213.   Exercise. 

5  —  ox  _Sx  —  9 
2  '~3~~' 

ISoLiTioN.  — The  1.  c.  d.  is  0,  and  we  multiply  through  by  it. 
o—'3x        8  ./  —  9 


2      —      :} 

lo  —  \)x  =  Ujx  —  lS 

_().,._  10.x- =  -15  —  18 

—  25.r  =  —33 

2. 

X       XX       x-\-2 
2  "^  3       4  "      2     ' 

3. 

x-1       x—3       x—3 

2               4      ~      2     ' 

4. 

5  =  8-^. 

X 

5.  •:  -  I-:,' =  42,-v 


./■         ./•         ./•         X         X        X    ,     ^^  ^ 


;5      4 

""      4 


7.    ^■_:!l:i_^.    •'■  +  •'      - 


-'+:=i(^-E)-:;+3(^^-i} 


9. 

'+'s  =  '~^- 

10. 

^  ..  _  .;}(*,       A)x  —  .18 
.2                .9 

11. 

ax+l,  =  l  +  l. 

12. 

X  —  a       X  —  b 

b  —  X  ~  a  —  X 

1Q 

,.       ,       4          , 

(Hyp-) 

(Ax.  3.) 

(Ax.  2) 

(86) 

(Ax.  4) 


X  +  2    '   X  +  (i 


SIMI'LK    K(,H  ArioNS.  IIK) 

214     Examples  of  the  Complete  Solution  of  Equations. 

11       ^        13  1' 

Mui»KL  Sun  HON.  — 1.  c.  (I.  =L'X  11X13. 

182  .c  -  208  H-  330  j:  -f  IKi  =  s:>8  j?  -  4433  +  143  or  (Ax.  3) 
182a-  -f  330a:  -  Ho>^  j-  —  143  j'  =  2(»8  -  17G  -  44;i3  (Ax.  a) 

(86) 
(Ax.  4) 


-489a:  =-44 

101 

a:=y. 

rKRIFH  ATIOX. 

7x9-8      ir)X04-8_  ..  ^, 

11          '           13 

.-■■"-•i-^ 

i.e.,          r,  4-  11  -  27  - 

11, 

or,                     K;  =  IG. 

o      ^  +  4   _  a:-h5 

(55,  «) 


3. 


;^.r_8       3a' -7 
3(>        4r> 


a 

K 


"^     ^("-^^-i^("-'^=5^"-'^   + 


48 


Caitioxs.  — The  student  should  be  careful  not  to  make  mistakes 
in  addition,  fnultiplication,  fractions.  sii;ns.  etc.,  and  then  attribute 
the  failure  to  get  correct  results  lo  som.  tiling  unknown  about  equa- 
tions. We  will  refer  to  common  .sounts  of  error  by  articles  which 
the  student  would  do  well  to  look  up  and  fix  in  mind.  Reduce  com- 
plex fractions  to  simple  ones  (182),  and  i)erform  all  indicated  opera- 
tions either  before  or  after  clearing  as  may  be  found  convenient. 
See  100,  4  ;  168,  a  \  and  176.  Study  carefully  171  «,  and  its  applica- 
tion to  e(juation8.  Reduce  answers  to  their  lowest  tenns  (169).  An 
answer  may  have  to  be  changed  to  make  it  identical  with  that  given 
(see  167).  For  factors  in  parentheses,  see  111,  2.  In  clearing  of  frac- 
tions remember  to  multiply  tlie  integral  (piantities  by  the  1.  c.  d. 
Other  references  might  be  given,  but  these  will  suggest  such  mistakes 
as  are  liable  to  hv  made. 


194  TEXT-BOOK   OF   ALGEBRA. 

215.  General  Rules  for  Solving  Simple  Equations  with  One 
Unknown. 

1.  Normal  process. 

(1.    First  clear  the  equation  of  fractions.    Theorem  212* 

(2.  Next  transpose  all  the  terms  containing  x  to  the 
left  member,  and  all  the  known  terms  to  the  right  member 
of  the  equation.     Theorem,  210. 

(3.  Then  by  87,  4,  collect  the  terms  on  both  sides,  and 
when  necessary  form  the  coefficient  of  x  as  in  97. 

(4.  Finally,  use  axiom  4  to  find  the  value  of  the 
unknown. 

2.  General  process. 

(1.  Proceed  in  any  way  that  seems  advantageous  by 
use  of  the  axioms  (207)  to  reduce  the  equation  to  the  form 
ax  =  h. 

(2.    Finish  the  solution  as  in  208. 

Rf:mark.  — The  first  rule  is  straightforward  in  its  method,  and 
so  best  for  the  beginner.  The  second  is  superior  to  the  other  for  the 
expert  student.  It  may  be  emphasized  here  that  any  process  in 
accordance  with  the  axioms  will  always  lead  to  correct  results ;  and 
any  jrrocess  or  operation  not  in  accordance  with  them  will  lead  inevi- 
tably to  incorrect  results. 

216.  Find  the  Value  of  the  Unknown  Quantity  in  the  fol- 
lowing, and  verify  the  answers  by  substituting  them  in  the 
original  equation  (189,  a). 

1.  l|.=8^.  3.|x  +  12  =  |x  +  6. 

2.  ^1+23^9  4    l-"_4  =  5. 

X  —  1  .  ^' 

7x-\-U       17 -3a;       4.^  +  2 
5.    5  -  6x  +  ^3—  =  —^ ^-. 


8IMPLK   EQUATIONS.  195 

9.    T)  X  -  (8  X  -  3  [K;  _  (i  ^ •  +  (4  -  5  xj]}  =  6. 
10.    «  (x  -  2)  +  2  .r  =  G  +  «. 


"■K"-"0-.'('-y+; 


lA. 


,  =  0. 


Solution.  —  The  1.  o.  d.  of  cocni.i.'iiis  .■  juils  12. 


3rt 

=  0 

(111,  2) 

5  X  =  2  a  —  a  +  —  f/  =  —  a 
5          5 

(Ax.  a,  174) 

ICATION. 

(Ax.  4) 

^  f  ?^  _  1!^  _   ^  f  ?^  _ -^    4.  1  f  —  _  -^  =  0 

2\2r>       3/       aV^o       •*/        4\25       5/ 
l/_jfir_\_l/7^\       l/3a\^2^_7i^      9ja  ^ 
2\      75/       ;H100/"'' 4\2r,  /       300      300  "^  300        "• 

NoTK.  — In  some  literal  eciuations  verification  is  easy,  in  others  it 
is  more  or  less  difticult.  In  the  following  examples  the  verification 
Is  rather  complex,  though  of  course  it  can  always  be  perfonned.  In 
numerical  problems  verification  is  always  comparatively  easy. 

12.  {it  H-  a-)  (/>  -f  ^)  =  (w  +  x)  {ii  +  X). 

13.  ^  -  -        =  ^    _ 

p  r        7 


196  TEXT-BOOK   OF   ALGEBRA. 


3a;  ,  7         1 
"^^-      4+0        7x  = 

5 
"6x  ' 

4 

7 

7         1    _    5        4 
6        7x      6x      7 

49  a;  -  6  =  35  -  24  X 

49  a;  +  24  X  =  35  +  6. 

41 
x  =  — .     Ans. 

.  3x 
+  -J-  •  (163, 159) 

(Ax.  2) 

(Ax.  3) 
(Ax.  1) 

(Ax.  4) 


15.  8  (a;  -  1)  + 17  (x  -  3)  =  4  (4a;  -  9)  +  4. 

a  c 

16.    


a  —  x 


X  —  1        4x  -  ^        7  X-  6       ^       X  —  2       3  a;— 9 
3      +        5         ~       8^  ^"^  ~Y~  +     10 
18.    Q>hx  -\-A.a''  -\-2ax  =  %  a%x  -  3  ax. 
,n      3a;  +  l        x-2 


3(a;— 2)  ~  a;  -  1 

X  —  7       2  a;  —  15  1 


x-\-l  ~  2a;  -  6        2  (a;  +  7) 
4  7  37 


21.    4- 

o;  +  2  ^  a;  +  3  ~  ar'  +  5  a-  +  6 

22.  7      _6a^  +  l_3(l  +  2^^) 
'a;  —  1         a;H-l  a;-  —  1 

a!-3      _     x-^  1^ 

^^'     4  (a;  -  1)  -  6  (a;  -  1)  +  9  ' 

1  a  —  h  1  a  -\-  b 

a  —  0     ^        X  a  -\-b    ^       x 

a  —  b      a  +  b  1  1 


Solution. 


X  X  a  +  b      a  —  b 

a  —  b  —  a  —  b       a  —  b  —  a  —  b 


X  cfi  —  62 

2  6        —  2  6 


X  a^ 

1  1 


(Ax. 

'^) 

(176)> 

(Ax. 

4) 

(Ax. 

3) 

X  «2  _  ^2 

X  =  a^  —  /A     Ans. 

This  solution  is  shorter  and  easier  than  if  the  equation  had  first 
been  cleared  of  fractions  (215,  2). 


SIMPLK   KQITATIOXS.  197 

25.  ax-\-bx  _      __      1       _  1 

b  a  —  b       a 

SuooESTiox.  —  Add  the  nu'iiibers  of  the  equation  as  they  stand 
before  clearing. 

26.  {X  -  b)  (X  -2)  -(x-^o)  (2x  -  r>)  +  (0-  -h  7)  (x-  2) 

=  0. 

27.  >  (L'x  -  10)  -  ^  Qix  —  40)  =  lb  -  ^  (57  -  x). 

28.  3i-;L>8-g  +  l>4);-=;H{2i+f}- 

29.  (a  ■i-x)(b-\-x)-  a  (^  +  c)  =  --  +  x\ 

b 

30.  (j'-h  1)'  =  S^)  -  (I  -x)\x-  2. 

31.  .5  a:  —  .3  ar  =  .25  X  —  1. 

32.  4.8^-  — .72  a-  — .05 


=  1.6ar  +  8.9. 


.5 
«3    1         3  ^1-x 

34.  ''•7'  +  -^^  =  .ooia.  +  .(;---2 

5  .5  .05 

36.  1  ■  1  1 


36. 


«i  —  ay        be  —  b)/        ac  —  ay 
1111 


X  —  2        a*  —  4        .r  —  ()        .r  —  8 

SuooKSTiON.  —  Add  the  members  (176). 

„     ax^  -\-  bx  -\-  c       ax  -{'  b 

01 ,    — J = . 

px^  -\-  qx  -\-r      px  -}-  q 

38.    ax-  —  hx  =  hx^  —  ex. 

Suggestion.  —  Divide  through  by  x. 

39    0a?4-3   ,  3x-6^2      3a'  +  22 
27      ■^2x-5      3"^        9 


198  TEXT-BOOK    OF    ALGEBIIA. 

Solution.  — Clearing  of  monomial  denominators  only  (215,  2), 

81  x  -  102 
9aj  +  3-f  -^   .    -      .    .- 


2     _  r  "  —  Ao  T  y  X  -r  oo 

li^x.  .3; 

81  X  -  102  _  ^^ 
2x-5 

(Ax.  2,  86) 

81  X  -  102  =  102  X  -  405 

(Ax.  3) 

-  81  X  =  -  243 

(Ax,  a) 

x  =  3. 

(Ax.  4) 

40    ^  (^  ^  -  3)        11  .T  -  1  _  9  g;  + 11 

14  +    3  0.'  +  1  ~  ~~7 


.^7x  —  6          cc  —  5           .T 

85           6;r-l()l~5 

4^    6-5x        7-2x-^         1  +  3.T       10:r 
15          14(.r-l)             21 

30 

^105 

^2    cc  —  l       X  —  2  _x  —  0       ic  —  6 

a'-2       ic  — 3       a;  — 6       x-T 

Solution. 

l+,_,      1       ,_3-l+^_,       1-     ^_, 

(161) 

1111 

X  —  2       X  —  3~x  —  0       X  —  7 

-1                             -1 

(176) 

(x_2)(x-3)-(x-e)(x-7) 

-x2+13x-42  =  -x2  +  5x-6 

(Ax.  3) 

8  X  =  30 

(Ax.  0,  86) 

x  =  4i 

(Ax.  4) 

44.    Solve  br  =  p  regarding,  first  h,  then  r,  and  lastly  />  as 
the  unknown  quantity. 

First,  Second,  Third, 

hr=^p  (Hyp.)         hr  =  p  (Hyp.)         ^'r  =  p  (Hyp.) 

Z*  =  1^  (Ax.  4)  r  =1  (Ax.  4)  'or  7^  =  hr. 


si.Mi'ij-:  K(^(Arn »Ns.  199 

This  equation  hr  =jt>  is  the  fuiuhiinental  formula  in  Pei- 
eentage  in  arithmetic,  b  standing  lor  kise,  r  for  rate  ex- 
}»resse(l  (h'cimally,  and  j)  for  the  percentage.      The   two 

equations   obtained,  b  =  J—^  and  /•  =  —  ,  are  tiie  formulae 

r  p 

for  the   cases,  given  the   percentage  and  rate  to  lind  the 

base,  and  given  the  base  and  percentage  to  find  the  rate. 

45.  Solve  a  =  b  (1  -\-  r)  for  ^,  and  r.     (a  =  "  amount ''). 

46.  Solve  d  —  b  (1  ~  /•)  for  b,  and  r.     {(l  =  '•  difference  "). 

47.  Solve  I  =  jjrn  for  jo,  r,  and  n  in  succession. 

By  the  definition  of  interest,  (/  =  interest,  p  =  principal, 
r  =  rate,  and  7i  =  time  in  years),  i  =  pm ;  we  are  to  find 
the  principal  when  the  rate,  time,  and  interest  are  given ; 
next,  to  find  the  rate,  when  the  principal,  time,  and  interest 
are  given ;  lastly,  to  find  the  time  when  the  principal,  rate, 
and  interest  are  given. 

48.  (1)  Solve  f  =  i  for  «  and  c.     (2)  Solve  t  ==  1  for  e. 

c  c 

49.  Solve  — —  -|- =  0  for  rt,  i,  c  and  d  in  succession. 

b  —  c      b^  d 

50.  Solve  ^  -f  1  =  0  in  the  same  manner. 

cd 

217.  General  Remarks  on  the  Solution  of  Equations. —  By 
Kule  '2,  the  student  i.s  at  liberty  to  pursue  aiiij  course  in  con- 
sonance with  axiomatic  principles.  Certain  of  the  exam- 
jdes  of  the  preceding  article  have  exhibited  an  advantage 
gained  by  deviating  from  the  normal  i)rocess.  The  follow- 
ing precepts  have  already  been  exemplified  in  one  or  another 
of  the  exercises. 

1.  Mark  at  the  outset  and  after  every  reduction  whether 
the  equation  can  be  divided  tlirough  by  any  factor,  mono- 
mial or  other,  and  remove  it.     (Axiom  4.) 


200  TEXT-BOOK   OF   ALGEBRA. 

2.  Study  the  original  equation  and  each  derived  equation 
to  see  if  by  uniting  certain  terms  as  they  stand,  or  after 
transposition,  quantities  can  be  made  to  disappear,  and  the 
equation  be  thereby  simplified. 

3.  Examine  whether  any  simple  reduction,  as  that  of  an 
improper  fraction  to  a  mixed  number,  will  pave  the  Avay  for 
the  simplification  suggested  in  2. 

4.  Consider,  before  applying  axiom  3,  whether  it  would 
not  be  better  to  clear  the  equation  of  only  a  part  of  its  de- 
nominators, and  then  simplify,  clearing  of  the  remaining 
denominators  afterwards. 

SECTION  III. 

•  Problems  Involving  Slmple  Equations  containing  One 
Unknown. 

218.  The  Problems  of  Algebra  (see  47  and  85)  are  quite 
like  those  of  arithmetic,  except  that  the  former  are  usually 
much  more  difficult ;  but  the  algebraic  treatment  of  them 
is  different.  In  arithmetic  we  were  accustomed  to  start 
with  what  was  given  and  work  towards  the  result ;  while 
in  algebra  we  conceive  the  problem  to  be  solved,  and  pro- 
ceed as  if  we  were  verifying  our  answer,  using  meanwhile 
a  letter  to  stand  for  it,  since  we  do  not  know  its  value. 
Going  through  the  form  of  a  verification,  the  letter  used 
everywhere  taking  the  place  of  the  unknown  number,  an 
equation  results,  the  solution  of  which  gives  the  value  of 
the  letter,  and  solves  the  problem.  This  is  called  the 
analytic  method. 

219.  The  Solution  of  a  Problem  involves  two  operations. 
1.    The  Statement  of  tho  Prohlem. — The  statement  of  a 

problem  is  its  expression  in  algebraic  symbols  in  the  form 
of  an  equation. 


siMi'r>K  K(.)r.\Ti(>\s.  -201 

2.  The  Solution  of  the  Kqiidtion.  —  Section  II.  of  tins 
clijipter  dealt  with  the  systematic  solution  of  equations,  and 
the  student  is  supix)sed  to  be  familiar  with  this  part  of  the 
in'ooess. 

220.  In  the  Enunciation  of  every  problem  we  notice  two 
things.     (See  those  in  47  and  85.) 

1.  The  description  of  an  unknown  number  (or  quantity) 
together  with  one  or  more  others  dependent  upon  the  first 
for  their  values. 

2.  The  a.ssertion  that  two  numl)ers  or  quantities,  obtained 
in  different  ways,  are  equal. 

The  rule  to  be  given  explains  the  translation  of  the  prob- 
lem into  algebraic  language.  It  will  be  convenient  before 
giving  the  rule  to  define  the  term  ^'function  of  a  quantity." 

221.  A  Function  of  a  Quantity  is  one  which  depends  upon 
it  for  its  value. 

Thus,  '^x  -\-  i)  is  a  function  oi  x.  If  a-  =  6,  then  the 
function  equals  3  X  (>  +  5  =  2.S ;  if  a;  =  9,  the  function 
equals  3X94-'^  =  32,  and  so  on.  From  this  we  can  see 
that  any  quantity  which  contains  x  is.  a  function  of  x,  and 
any  quantity  which  contains  a  is  a  function  of  r^  and  so  on. 

222.  General  Rule  for  Solving  Problems.  —  Compare  with 
the  solutions  of  Arts.  47  and  85. 

1.  Let  a?,  or  some  convenient  function  of  x,  represent  the 
unknown  cpiantity. 

2.  Express  the  different  functions  of  x  to  be  used  in 
forming  the  equation  in  terms  of  x. 

3.  Construct  the  two  expressions  described  as  equal,  and 
make  them  the  two  members  of  an  equation. 

4.  Solve  the  equation. 

5.  Obtain  the  values  of  such  functions  of  the  root  as  the 
[)roblem  asks  for. 

a.  Illustkatioxs.  —  In  the  examples  as  set  down  in  47  and  85 
we  first  wrote  x  for  the  unknown  quantity;  then  underneatli  iU  func- 


202  TEXT-BOOK  OF  ALGEBRA. 

tions.  Afterwards  the  equation  was  formed,  which  was  solved  ac- 
cording to  the  best  light  we  then  had.  To  particularize,  let  us  take 
Ex.  5,  Art.  47.  The  functions  of  x  are  2  x  and  2  x  +  7.  The  equation 
is  then  formed,  after  which  it  is  solved,  giving  x  =  8.  But  the  prob- 
lem requires  two  other  answers,  viz.,  2  x  and  2  x  +  7,  which  are 
found  equal  to  16  and  23  respectively. 

h.  The  two  sides  of  an  equation  must  be  expressed  in  terms  of  the 
same  unit.  Of  course,  one  side  cannot  be  expressed  as  a  certain 
number  of  dollars,  and  the  other  as  an  equal  number  of  cents, 
neither  can  one  side  denote  metres  and  the  other  centimetres,  and 
so  on.  Careless  thinking  sometimes  leads  a  student  even  into  such 
absurdities  as  these. 

223.  Proportions  are  Transformed  into  Equations  by  means 
of  the  fundamental  property  tliat  the  product  of  the  means 
equals  the  product  of  the  extremes. 

224.  Exercise  in  the  Solution  of  Problems  in  One  Unknown 
Quantity. 

1.  A  gentleman  divides  two  dollars  among  twelve  chil- 
dren, giving  to  some  18  cts.  each,  and  to  the  others  14  cts. 
"each.     How  many  were  there  of  each  class  ? 

Let  X  =  the  number  receiving  18  cts.  each,  (222,  1)^ 

Then  12  —  a?  =  the  number  receiving  14  cts.  each,  (222,  2) 
18  0?  =  the  number  of  cents  the  first  class  received, 

(222,  2) 
(1^  —  ^)  14  =  the  number  of  cents  the  second  class  re- 
ceived, (222,  2) 
18  X  +  (12  -  x)  14  =  200,                      (222,  3,.  and  b) 
18 a^  +  168  -Ux  =  200  (110) 
4:X  =  200  -  168  =  32                      (Ax.  2) 
x  =  S    I                                             (Ax.  4) 
12-^  =  4    \                                         (222,5) 
Verification.     8  X  18  +  4  X  14  =  144  +  56  =  200. 

2.  There  are  three  consecutive  numbers  such  that  if  they 
be  divided  by  10,  17,  and  26,  respectively,  the  sum  of  the 
quotients  will  be  10;  find  them. 


Sl.MI'J.h     i.i,'(    A  i  loN.s.  20^ 

Here  the  student  may  liave  trouble  in  expressing  the  three  num- 
bers (220,  1).  The  U\v<i  is  tliat  the  three  numluTs  follow  one  an- 
other, the  second  being  one  greater  than  the  tirst,  and  the  third  oiw 
greater  than  the  second. 

Let  a;  =  the  least  number,  (222,  1) 

tluMi     X  +  1  =  the  next  greater,  (222,  2) 

X  +  2  =  the  greatest  number,  (222,  2) 

■^  +  ^+^=10  <222,:i) 

442  X  +  260  X  +  200  +  170  X  +  340  =  44200,  (A x.  :J) 

872  X  =  44200  -  200  -  340  =  43000,  (Ax.  2) 

a!  =  50  ]  (Ax.  4) 

x  +  l=r,l    I  (222,5) 

X  +  2  =  52  J  (222,  .-)) 

50       51        52 
Verification.    IL.^1 — l  _  =  -  _l  •>  _|_  o  _  ia 

3.  All  estate  is  divided  among  four  children  in  sutdi  a 
manner  that 

the  first  has  $200,  more  than  ^  of  the  whole, 
"    second      340,      ''        "     ^     "  " 

**    third         300,      "        '*     ^     " 
"    fourtli       400,      "         ''    i     " 
what  is  the  value  of  the  estate  ? 

Here  the  beginner  may  not  see  how  the  statement  of  220,  2  is  true 
of  tliis  problem.  It  must  be  taken  rather  by  implication  than  by 
the  direct  statement  of  the  problem. 

Let  X  =  the  value  of  the  estate  (222,  1) 

^+  200  =  the  number  of  dollars  of  the  first  child's  port  ion,  (222,  2) 
4 

^-1-340="         "        "       " 
5 

^-h:J0O  =  "        "        "      " 

-  +  400=" 
8 

|-+ 200 -I- 1+340 +1+300+1  + 

4-+!+l+^-^=-^-^^ 

30x  +  24x  +  20x  +  1.5 X  -  120x  =  -  148800 
_  :31  X  =  -  148800 
x  =  4800. 


second     " 

"       (222,  2) 

lliird        " 

"       (222,  2) 

fourth      " 

-      (222.  2) 

=  x 

(222,  :J) 

204  TEXT-BOOK   OF   ALGEBRA. 

Verification. 

4800  .      4800       4800       4800 

-^—  +  200  +  -^  +  340  +  — ^  +  300  +  -^  +  400  =  4800. 

4  D  O  O 

4.  A  boy  bought  an  equal  number  of  apples,  lemons,  and 
oranges  for  56  cts. ;  for  the  apples  he  gave  1  ct.  apiece,  for 
the  lemons,  2  cts.  apiece,  and  for  the  oranges  5  cts.  apiece. 
How  many  of  each  did  he  purchase  ? 

5.  Divide  the  number  181  into  two  such  parts  that  4 
times  the  greater  may  exceed  5  times  the  less  by  49. 

6.  A  boy  ate  I  of  his  plums  and  gave  away  ^  of  them. 
The  difference  between  the  number  he  ate  and  the  number 
he  gave  away  was  3.     How  many  had  he  ? 

7.  What  number  is  that  whose  half,  third,  and  fourth 
parts  together  equal  63  ?     Verify  the  answer. 

8.  A  father  aged  54  years  has  a  son  aged  9  years  :  in  how 
many  years  will  the  age  of  the  father  be  just  4  times  that 
of  the  son  ? 

9.  A  father  is  now  40  years  of  age  and  his  daughter  13. 
How  many  years  ago  was  the  father's  age  10  times  that  of 
the  daughter's  ? 

10.  A  man's  age  and  his  wife's  age  now  bear  to  each 
other  the  ratio  of  6:5.  But  15  years  hence  they  will  be 
to  each  other  as  9  :  8.     How  old  are  they  now  ? 

5  X 
In  this  problem  we  might  let  x  =  the  man's  age,  and  — phis  wife's 

age  ;  however,  we  can  avoid  fractions  in  the  folloVing  preferable 

way  : 

Let  Gx  =  the  man's  age,  (222,  1) 

then    5  x  =  the  wife's  age,  (222,  1) 

a  X  +  V)  =  the  man's  age  15  years  later,  (222,  2) 

6x  +  15  =  the  wife's  age  15  years  later,  (232,  2) 
6x  +  15  :5x+  15  ::9  :8 

48  cc  +  120  =  45  X  +  135  (223) 


15:25  +  15  ::9:8. 


X  =    5 

633  =  30 
5x  =  25 

1^ 

ERIFICATIOX. 

30- 

SIMTLK    KgiATlONS.  205 

11.  A  gentleman  is  now  twenty-live  yeai*s  old,  and  his 
yoi^ngest  brother  is  15.  How  many  years  must  elapse 
before  their  ages  will  be  as  5  :  4  / 

12.  The  ditferenc;.^  between  two  numbers  is  V2,  and 
the  greater  is  to  the  less  as  11:5.  What  are  the  num- 
bers ? 

13.  What  two  numbers  are  as  3 :  4,  to  each  of  which  if 
4  be  added  the  sums  will  be  to  each  other  as  5  :  0  ? 

14.  A  farm  contains  26  acres.  Three  times  A's  part  is  6 
acres  less  than  4  times  IVs  part.  How  many  acres  had 
each? 

15.  Find  two  numbers  differing  by  10  whose  sum  is  equal 
to  twice  their  difference. 

16.  A  man  has  four  children  the  sum  of  whose  ages  is 
48  years,  and  the  common  difference  of  their  ages  is  ecjual 
to  twice  that  of  the  youngest.     Find  their  ages. 

17.  Three  boys  are  talking  of  their  money.  A  says  to  B, 
''  I  have  three  times  as  many  cents  as  you  have."  Says  C 
to  A  and  1^,  *'  I  have  as  much  as  the  difference  between 
your  sums.'-  Now,  A's  money  achled  to  twic(^  IVs  and  twice 
C's  makes  03  cents.     How  much  had  each  ? 

18.  A  grocer  has  two  sorts  of  sugar,  one  wortli  *.)  cents. 
and  the  other  13  cents  a  i)ound.  How  many  jHuuids  of  eacli 
sort  must  l>e  taken  to  make  a  mixture  of  a  liundrcd  ])()unds 
worth  VJ  cents  j)er  jxjund  *.' 

19.  \  ]>ayniastcr  wisliing  to  draw  for  use  on  pay  day  the 
sum  (jf  .il»113<)()  recpiested  tlic  teller  to  m.ike  up  the  sun* 
with  l)ank-l)il's  of  different  denominations  as  f(jllows :  a 
certain  number  of  lOO's,  twice  as  many  5()'s,  twice  as  many 
-O's  as  5(Vs.  twice  as  many  tlTs  as  2()*s.  twice  as  many  5';s 
a,s  lO's,  twice  as  numy  l''s  as  5's,  and  twice  as  many  I's  as 
-■-      How  nianv  of  each  denomination  were  needed  V 


206  TEXT-BOOK  OF  ALGEBRA. 

20.  A  grocer  bought  two  casks  full  of  oil,  one  of  which 
held  twice  as  much  as  the  other.  I'roui  one  cask  he  drew 
10  gal.  and  from  the  other  1  gal.,  and  then  drew  from  the 
larger  i  of  the  remaining  oil,  and  found  that  the  two  casks 
now  contained  equal  quantities  of  oil.  How  much  did  each 
hold  when  full  ? 

21.  The  garrison  of  a  certain  town  consisted  of  125  men, 
partly  cavalry  and  partly  infantry.  The  monthly  pay  of  a 
cavalryman  is  $20,  and  that  of  an  infantryman  is  $15,  and 
the  whole  garrison  receives  $2050  a  month.  What  is  the 
number  of  cavalry  and  what  of  infantry  ? 

22.  Divide  88  dollars  between  A,  B,  and  C,  giving  to  B  f , 
and  to  C  f  as  much  as  to  A. 

23.  A  man  having  completed  f  of  his  journey,  finds  that 
after  traveling  30  miles  farther  only  f  of  the  journey 
remains,     liequired  the  length  of  the  journey. 

24.  Two-fifths  of  a  pole  is  in  the  water,  one-tenth  in  the 
mud,  and  the  remainder  out  of  the  water.  There  are  6  ft. 
less  of  it  in  the  water  than  in  the  mud  and  above  water. 
How  long  is  the  pole  ? 

25.  There  are  two  silver  cui)s  and  one  cover  for  both. 
The  first  weighs  12  oz.,  and  with  the  cover  twice  as  much  as 
the  other  without  it;  but  the  second  with  the  cover  weighs 
1  more  than  the  first  without  it.  Find  the  weight  of  the 
cover. 

26.  I  have  a  certain  number  in  mind.  I  multiply  it  by  7, 
add  3  to  the  product,  and  divide  the  sum  by  2  ;  I  then  find 
tliat  if  I  subtract  4  from  the  quotient  I  get  15.  What  num- 
ber am  I  thinking  of  ? 

27.  A  certain  number  multiplied  by  5,  24  subtracted 
from  the  product,  the  remainder  divided  by  6,  and  this  quo- 
tient increased  by  13,  results  in  the  number  itself.  What 
is  the  number  ? 


simimj:  I  ..•!   \  I  loNs.  '201 

28.  A  |)erson  spent  \  of  his  moiu'y,  alter  which  he  earned 
$'.i.  Then  he  lost  .\  of  wliat  he  then  had,  receiving  after- 
wards $2  back.  Lastly  he  gave  away  }  of  what  he  now 
had,  when  he  found  he  had  only  $12  remaining.  "What 
had  he  at  first  ? 

Lot  .r  =  the  number  of  dollars  lie  Imd  at  first,  (222,  1) 

tlun     -  J"  =  remainder  after  spending  one  fourth,  (222,  2) 

4 

2  /3  \ 

.y  I  7-  X  +  ;H  =  the  second  remainder,  (222,  2) 

!!  r|  r^  J-  4-  :J  j  +  21  =  the  third  remainder  =^  12.  (222,  :i) 

2l(^^^)+7-^^  ao9,.) 

7  +  7  +  7  "  ^^ 
3x+  12  +  12  =  84 
a:  =  20     Ana. 

Vkrificatiox.    i  of  20  =  15;  15  +  3  =  18;  |  of  18  =  12;  12  +  2 
=  14;  ?  of  14  =  12=  12. 

29.  A  basket  of  oranges  is  emptied  by  one  i)erson  taking 
half  of  them  and  one  more,  a  second  person  taking  half  of 
the  remainder  and  one  more,  and  a  third  j)erson  taking  half 
of  this  remainder  and  <*>  moi-c.  How  in;niy  did  the  basket 
contain  at  first. 

30.  In  making  a  journey  a  traveler  went  on  the  first  day 
^  of  the  distance  and  8  miles  more ;  on  the  second  day  he 
went  ^  of  the  distance  that  remained  and  15  miles  more  : 
and  tlie  tJiird  day  he  went  ^  of  the  distance  that  remained 
and  12  miles  more;  on  the  fourth. day  he  went  .S5  miles  ;nid 
finished  his  journey.     What  was  the  distance  traveled  . 

31.  A  man  who  has  $2(X)0  invested  in  a  mill  from  whicii 
he  receives  a  certtiin  per  cent,  and  $10(K)  in  real  estate  from 
which  he  derives  oidy  J  of  the  previous  rate,  has  an  income 
from  l)oth  of  $330.     What  rate  per  cent  does  he  receive  ? 


208  TEXT-BOOK   OF    ALGEBRA. 

Let  X  =  rate  %  on  the  $2000  investment, 
'^  X  =  rate  on  the  $1000  investment  ; 

.-.  2000  X  ^  =  20x  =  the   interest  on  the   first,   and 

ivA' 


1000  x-i^ - 
100 

30  X 
■    4 

=  interest  on 

rate  per 

cent. 

20  X 

30  ic 

"*"    4     " 

=  330 

80  X 

+  30x  = 

:  1320 

110  x  = 

1320 

. 

X  = 
ix  = 

12%  1 
9%) 

32.  What  is  the  property  of  a  person  whose  income  is 
$430,  when  he  has  §  of  it  invested  atAfc,  i  SitSfc,  and  the 
remainder  at  2^. 

33.  A  person  who  is  worth  $12000  invests  a  certain 
amount  in  railroad  stock  from  which  he  receives  6^.  One- 
third  of  the  diiference  pays  4^  interest,  and  the  other  two- 
thirds  5fo  interest,  and  from  all  his  funds  he  clears  $896. 
What  amount  was  invested  in  railroad  stock  ?  On  what 
amount  had  he  realized  but  $600  ?     (See  252,  7.) 

34.  A  man  has  $5000  invested  at  a  certain  rate,  $2000  at 
double  the  former  rate,  and  $1000  at  triple  the  first  rate. 
He  received  from  a  property  an  income  of  $1900,  but  pays 
out  $2000  for  improvements,  personal  expenses,  insurance, 
etc.  He  finds  he  has  $500  remaining  at  the  end  of  the 
;[ear.     What  does  he  receive  on  the  $2000  ? 

35.  A  workman  engaged  for  48  days  at  the  rate  of  $2 
l)er  day  and  his  board.  But  for  every  day  he  might  be  idle 
he  was  to  pay  $1  for  his  board.  At  the  end  of  the  time  he 
received  only  $42.     How  many  days  did  he  work '.' 

36.  A  man  hired  a  laborer  for  one  year  at  the  wages  of 
$90  and  a  suit  of  clothes.  At  the  end  of  7  months  the 
laborer  quit  his  service  and  received  $83.75.  At  what 
price  were  the  clothes  estimated  ? 


S1M1MJ-:  i:(,>i  A  rioNs.  209 

37.  A  boy  engaged  to  carry  «'50  glass  vessels  to  a  cer- 
tain place  on  condition  of  receiving  o  cents  for  every  one 
he  delivered  safe,  and  f<  rh  itini;  12  cents  for  every  one  he 
should  break.  On  settlement  he  received  99  cents.  How 
many  did  he  break  ? 

38.  A  tree  standing  vertically  on  level  ground  is  60  feet 
high.  Upon  being  broken  over  in  a  storm  tlie  upper  part 
reached  from  the  top  of  the  trunk  to  the  ground  just  30  feet 
from  the  foot  of  the  trunk.  What  was  the  length  of  the 
part  broken  off  ? 

Note.  —  This  problem  tlei>ends  upon  the  theorem  proved  in 
geometry  and  stated  in  most  arithmetics,  that  the  square  on  the 
longest  side  of  a  right-angled  triangle  is  equal  to  the  sum  of  the 
squares  on  the  other  two  sides. 

39.  A  can  perform  a  i)iece  of  work  in  ()  days,  B  can  per- 
form the  same  work  in  8  days,  and  C  in  24  days.  In  how 
many  days  can  they  finish  it  if  all  work  together? 

Let  j;  =  the  number  of  days  when  all  work  together.  Now,  A  does 
j^  of  the  work  in  one  day,  and  in  x  days  he  will  do  x  times  jl  or  ^  of 
the  work.  For  a  like  reason  B  will  do  g,  and  C,  ^4  of  the  work  in 
one  day.     But  they  do  one  times  the  work  in  x  days,  therefore 

1+1+  2T=^  (219,1) 

.-.     x  =  3.*     Ans.  (219,1) 

Note.  —  The  whole  work,  as  is  customary  in  intellectual  arith- 
metic, is  called  one;  then  the  fractions  of  the  work  which  each  one 
does  added  together  make  the  whole  work,  or  one. 

TIk'  equation  i  H"  i  "I"  5V  '^^  x  (virtually  the  same  as  the  one  just 
given,  being  obtained  from  it  by  dividing  through  by  x,  Ax.  4)  is  found 
by  saying  that  if  it  takes  all  of  them  x  days,  in  one  day  they  will  do 
^  of  the  work.     Solving  this  equation  x  =  3  as  before. 

40.  A  and  ?>  together  can  do  a  piece  of  work  in  12  days, 
A  and  C  in  15  days,  and  B  and  C  in  20  days.  In  what  time 
can  they  do  it  all  working  together  ? 

>  Hereafter,  when  it  Is  cuiiveiiUnt  to  leave  the  solution  of  an  equation  to  the 
student,  the  answer  will  be  given  witli  a  refi  r<  iic<  to  219,  >'• 


210  TEXT-BOOK    OF    ALGEBKA. 

Let  X  =  the  number  of  days  in  which  all  can  do  the  work.  Then 
X  —  tV  ^^  the  part  C  can  do  in  one  day,  etc. 

41.  A  cistern  can  be  filled  by  one  pipe  in  15  minutes,  by 
another  in  12  minutes,  and  by  a  third  in  10  minutes.  In 
what  time  can  it  be  filled  if  all  are  left  open  ? 

42.  A  tank  can  be  filled  by  two  pipes  in  24  and  30  min- 
utes respectively,  and  emptied  by  a  third  in  20  minutes. 
What  time  will  it  be  in  filling  if  all  are  left  open  ?  • 

43.  A  man  who  can  perform  a  piece  of  work  in  14  days, 
works  4  days  when  he  calls  in  a  boy,  and  together  they  fin- 
ish the  job  in  6  days.  In  how  many  days  could  the  boy  do 
the  work  alone  ? 

44.  A  cistern  can  be  filled  by  two  pipes  in  9  minutes  and 
12  minutes,  and  emptied  by  two  others  in  15  and  18  minutes 
respectively.  The  first  pipe  is  left  open  one  minute,  then 
the  third  is  opened.  At  the  end  of  the  second  minute  the 
second  pipe  is  opened,  and  at  the  end  of  the  third  minute  the 
fourth  is  opened.  How  soon  will  the  cistern  be  in  filling 
counting  from  the  time  the  first  pipe  was  opened  ? 

45.  A  boy  buys  apples  at  the  rate  of  5  for  2  cts.  He 
sells  half  of  them  at  the  rate  of  2  for  1  ct.,  and  the  rest  at 
the  rate  of  3  for  1  ct.,  and  clears  1  ct.  by  the  transaction. 
How  many  did  he  buy  ? 

46.  A  market  woman  bought  some  eggs  at  the  rate  of  2 
for  1  ct.,  and  as  many  more  at  the  rate  of  3  for  1  ct.  She 
sold  them  all  at  the  rate  of  5  for  2  cts.,  and  found  she  had 
lost  4  cts.     How  many  of  each  sort  did  she  buy  ? 

47.  A  train  on  the  Northwestern  line  passes  from  Lon- 
don to  Birmingham  in  3  hours ;  a  train  on  the  Great 
Western  line,  which  is  15  miles  longer,  traveling  at  a 
speed  which  is  less  by  1  mile  per  hour,  passes  from  one 
place  to  the  other  in  3^  hours.  Find  the  length  of  each 
line. 


SIMl'LK    Kijl  AlloNS.  211 

Hemauk. — Problems  of  distance,  time,  and  rate  occur  so  fre- 
quently, that  it  may  be  helpful  to  the  student  to  be  familiar  with  the 
answers  to  the  following  gemial  (jih  stions  :  — 

Given  the  time  and  rate,  how  is  the  distance  found  ? 

Given  the  distance  and  rate,  how  is  the  time  found? 

Given  the  distance  and  time,  how  is  the  rate  found? 

Similarly  in  problems  involving  price,  number  of  articles,  and 
total  cost. 

Given  the  price  and  the  number  of  things,  how  is  the  total  cost 
found  ? 

Given  the  price  and  the  total  cost,  how  is  the  number  of  things 
found  ? 

Given  the  number  of  things  and  the  total  cost,  how  is  the  price 
found  ? 

So  also  we  might  discuss  problems  involving  length,  breadth,  and 
thickness,  area  and  solid  contents. 

The  student  must  decide  in  every  problem  what  kind  of  imits  the 
two  sides  of  the  equation  are  to  contain,  and  prepare  the  functions 
accordingly.     This  will  tend  to  make  the  process  of  solution  clear. 

48.  A  person  walks  to  the  top  of  a  mountain  at  the  rate 
of  2]\  miles  an  hour,  and  clown  the  same  way  at  the  rate  of 
.'U  miles  an  hour,  and  is  out  5  hours.  How  far  is  it  to  the 
top  of  the  mountain  ? 

49.  A  boy  who  runs  at  the  rate  of  12  feet  per  second, 
starts  20  yards  behind  another  whose  rate  is  10^  feet.  How 
socii  will  the  first  boy  be  10  yards  ahead  of  the  second  ? 

50.  The  distance  from  M.  to  L.  is  31^  miles.  The  express 
down  train  leaves  M.  at  11.30  a.m.,  and  arrives  in  L.  at 
1  -  ;<  >  P.M.  The  up  train  leaves  L.  at  11.45  a.m.,  and  arrives 
ui  M.  at  12.35  p.m.  Supposing  the  speed  of  each  train  to 
be  uniform,  when  will  they  meet  ? 

51.  A  bought  eggs  at  18  cts.  a  doz.  Had  he  bought  5 
more  for  the  same  money,  they  would  have  cost  him  2^  cts. 
a  dozen  less.     How  many  eggs  did  he  buy  ? 

52.  A  rows  4  miles  and  B  3  miles  an  hour.  A  is  14  miles 
farther  up  stream  than  B,  and  they  row  towards  each  other 


212  TEXT-BOOK    OF   ALGEBRA. 

till  they  meet  4  miles  above  B's  position.     How  rapid  is 
the  current  ? 

Suggestion.  —  Let  x  =  the  rate  of  the  current,  then  x  +  4  equals 
A's  rate  down,  3  —  ic  B's  rate  up. 

53.  A  boatman  who  can  row  6  miles  an  hour  in  still  water 
rows  a  certain  distance  down  stream  and  back  again  in  3 
hours.  How  far  did  he  row,  supposing  the  stream  to  flow 
3  miles  in  two  hours  ? 

54.  There  are  two  numbers  in  the  ratio  of  ^  to  f ,  which 
being  increased  respectively  by  6  and  5,  are  in  the  ratio  of 
f  to  ^.     What  are  the  numbers  ? 

55.  A  farmer  makes  a  mixture  of  rye,  oats,  and  barley, 
using  3  bushels  of  rye  as  often  as  4  of  oats,  and  5  of  barley. 
The  whole  amount  of  grain  used  was  66  bushels.  How 
many  bushels  were  there  of  each  ? 

56.  The  estate  of  a  bankrupt,  valued  at  $21,000,  is  to  be 
divided  among  4  creditors  proportionately  to  what  is  due 
them.  The  debts  due  A  and  B  are  as  2  and  3 ;  B's  and  C's 
claims  are  in  the  ratio  of  6:7;  and  D's  claim  is  equal 
to  A's.     What  sum  must  each  receive  ? 

SuGGESTiox.  —  Let  4  X  =  A's  and  6x=  B's  money. 

57.  A  general  arranging  his  men  in  the  form  of  a  solid 
square  finds  he  has  21  men  over,  but  attempting  to  add  one 
man  to  each  side  of  the  square  finds  he  wants  200  men  to 
fill  up  the  square.     How  many  men  had  he  ? 

58.  The  length  of  a  room  exceeds  its  breadth  by  8  feet. 
If  each  had  been  increased  by  2  feet,  the  area  would  have 
been  increased  by  60  square  feet.  Find  the  original  dimen- 
sions of  the  room. 


SIMPLK    K(^l  ATIONS.  21  ^J 


(  HAPTKK    XV. 

SIMPLi:     KQUATIONS    CONTAINING    TWO    OH    MOIJK     I'N- 
KNOWN    QUANTITIES.      ELIMINATION. 

225.  Simultaneous  Equations  are  sets  of  equations  which 
are  to  l)e  satisfied  by  the  same  values  of  the  unknown 
quantities  contained  in  tlieni. 

a.  All  proper  conditional  problems  (188)  give  rise  either  to  a 
single  equation  containing  one  unknown  quantity,  or  to  two  or 
more  equations  containing  as  many  unknown  (juantities.  In  the 
latter  case  some  of  the  funrtlous  (222,  2)  of  the  unknown  are  regarded 
as  other  unknown  quantitten.  Consequently  we  may  solve  problems 
in  one  of  two  ways  :  either  regard  the  problem  as  containing  but  one 
unknown  (x)  and  express  everything  in  terms  of  it,  as  we  did  in  the 
last  chapter  ;  or  regard  the  problem  as  containing  two  or  more 
unknowns  (x,  y,  z,  etc.),  and  form  two  or  more  equations.  To  make 
this  plain  let  us  take  Ex.  18,  224- 

1st.   With  one  unknown. 

Let  X  =  the  niunber  of  lbs.  of  1»  ct.  sugar,  (222,  1) 

then  100  —  ar  the  number  of  lbs.  of  1:]  ct.  sugar,  (222,  2) 

0  J-  +  13  (100  -x)  =  1200,  (219,  1) 

x  =  2r>    )  (219,2) 

100  -  .r  =  7.")    S 

2d.    With  two  unknowns. 

Let  X  =  the  numl>or  of  \hf.  of  0  ct.  sugar, 

and  y  =  the  number  of  lbs.  of  13  ct.  sugar, 

Equation  (1)  x  -\- y  =  100  )  /g^g  jv 

Equation  (2)  0  j  +  13  //  =  1200     S 

//  =  100-  X  (From  Eq.  (1)  by  Ax.  2) 

9  J'  -H  13  (100  -  J-)  =  1200  (Substituting  100  -  x  for  y  in  the 

second  equation) 
X  =  2r>  (219,  2) 

y  =  75  (By  substituting  25  for  x  in  (2)) 


214  TEXT-BOOK    OF    ALGEBRA. 

In  tlie  second  solution,  a  new  unknown  quantity,  ?/,  is  first  intro- 
duced and  then  removed  from  the  second  equation  by  substituting  its 
value  obtained  from  the  first.  Tlie  resulting  equation  is  the  same  as 
the  equation  of  the  first  solution.  In  this  problem  it  makes  little  dif- 
ference whether  we  solve  it  with  two  or  with  one  unknown  quantity; 
but  if  the  second  quantity  is  a  very  complex  function  of  the  first,  the 
second  method  has  an  advantage  over  the  first. 

h.  A  single  equation  containing  two  unknown  quantities  is  "  ?n- 
determinate,^''  since  the  value  of  either  of  the  unknowns  cannot  be 
found  until  that  of  the  other  is  known. 

To  illustrate  this,  suppose 

5  a;  -  3  2/  =  18 

DX  =  lS  +  Sy  (Ax.  1) 


_18  +  32/ 


o 


(Ax.  4) 


li  y  —  1,  then  x  =  ■ — ;: —  =  4^ 
lfy  =  2,  then  x  =  ^^  '^ ^  ""  ^  =  4* 

If  ?/  =  3,  then  x  =  — ^  =  5§  ^^^^^  ^^  ^^^ 

If,  however,  two  equations  are  given,  such  as 

(1)  Sx  +  4y  =  9 

(2)  2x  +  9y  =  11 

the  value  of  y  can  be  found  from  equation  (1) 

(1)  i^x  +  4y  =  9,4y  =  9-^;\x,y  =  '^-^^, 
and  then  placed  in  Eq.  (2),  obtaining 

2x  +  9   5^^     =17, 

and  we  have  resulting  one  equation  containing  but  one  unknown, 
which   solved,  gives  x  —  — ^ .     Similarly,  the  value  of  x  might  have 

been  found  from  (1)  and  substituted  in  (2),  whence  the  value  of  y 
might  have  been  found. 

c  Any  proper  axiomatic  operation  performed  on  an  equation  will 
not  change  the  value  of  any  letter,  and  so  will  not  change  the  "  iden- 
tity "  of  the  equation.  It  will  still  be  the  same  equation,  only  under 
a  new  guise.     (Consult  Art.  360.) 


SIMPLE   EQUATIONS.  215 

It  will  be  convenient  to  use  a  system  of  marking  equations  by 
which  it  can  be  indicated  that  a  certain  equation  appears  in  a  new 
fonn.  The  figures  written  at  the  left  in  parentheses  will  be  used  to 
designate  the  equations,  while  the  same  figures  with  subscripts  will 
distinguish  the  several  equations  in  their  new  forms. 
Thu3,  (1)  Sx  +  4y  =  2'l 

(2)   2j--3iy  =  21 

(1,)  9  X  +  12  y  =  81     [Eq.  (1)  multiplied  by  3,  Ax.  3] 
(2,)  8  x  -  13  2/  =  84     [Eq.  (2)  multiplied  by  4,  Ax.  3] 
(1.)      a;  +  1 2/  =  0        [Eq.  (1)  divided  by  3,  Ax.  4] 
and  so  on. 

226.  Elimination  is  the  process  of  deducing  from  two  or 
more  e(|Uiitions  containing  two  or  more  unknown  quantities 
a  single  equation  with  but  one  unknown  quantity. 

rt.  There  are  three  kinds  of  elimination  usually  given :  elimination 
by  substitution  ;  by  comparison  ;  and  by  addition  and  suhtrartion. 

We  shall  take  up  in  section  I.  the  case  in  which  there  are  but  two 
unknowns.  Then  in  the  next  section  that  in  which  there  are  three 
or  more  unknowns. 

SECTION  I. 

Solution  of  Simple  Equations  containing  Two  Unknown 

QlANTITIES. 

227.  Equations  Containing  Two  Unknown  Quantities  are 
solved  by  the  methods  of  elimination.  We  give  first  the 
methods  of  elimination  with  exercises,  and  then  problems 
involving  such  simultaneous  equations. 

I. -ELIMINATION  BY  SUBSTITUTION. 

228.  Elimination  by  Substitution  is  performed  by  finding 
the  value  of  one  of  the  unknowns  from  either  of  the  given 
equations  and  substitutin<^  it  in  the  other. 

229.  After  the  value  of  one  of  the  unknowns  has  been  found, 
it  is  substituted  in  one  of  the  foregoing  equations,  thus  giv- 
ing an  equation  conttiining  only  the  other  unknown  whose 
value  can  then  be  found. 


216 


TEXT-BOOK   OF   ALGEBRA. 


(1,)  8.T  =  68  — 5y 
68-5?/ 
8 


230.    Exercise  in  Elimination  by  Substitution. 

1.    Given     (1)  8.r +  o// =  68,  and  (2)  12a^  +  7?/  =  100 

(Ax.  2) 

(Ax.  4) 

(228) 

(178) 
(Ax.  3) 

(229j 


(1.)    X  -. 

(2)    Vl(^^^^^\  +  ^y  =  im 

204 -15^,^         _,.^ 
2 —    +  <  y  =  100  ^ 

204-  15ij  +  Uy  =  2()() 

y  =  4     J^is. 

(1)    8;r  +  5x4  =  68 
S  ^  =  48 
X  =  (J.     Ans. 


2.  Given 


(1) 


1; 


3^  _  3j/^-  4  :r  +  7      5 

To  ~  r>        ~^ 


^"^        3      ■+■  2  ~  20  ""  -^  -     15     +6+10 


(li)   4  v/  -  9  ci"  =  18  y  -  24  X  +  42  -  25  (Ax.  3) 

(1.)  15  ic  —  14  y  =  17  (Ax.  a,  and  86) 

15  X  —  17 


(la) 


// 


14 


(Ax's  n,  and  4) 


(2i)   20  y  -  20  +  30  .T  -  9  y  -  60  =  4  ?/  —  4  .r  +  10  .i-  -|-  (> 

(Ax.  3) 
(2^)  i>4  .r  -[-  7  y  =  86 

15  .T  -17' 


(2.) 


24  .r  +  7 


14 


86 


48.'r  +  15.r  —  17  =  172 
(',3  X  =  189 
X  =  3     A71S. 

(1,)  y  =  lA><^|::il^  =  2.    ^,«. 


(228) 


(229) 


siMi'Li-:  i:()r.\'ri(>Ns. 


217 


XoTE.  — As  either  unknown  may  Ix-  eliminated  the  student  is  left 
to  discover  which  one  it  will  be  advisable  to  eliminate  first.  As  a 
rule  that  unknown  should  beelimiiiated  which  has  the  bust  coefficient. 
In  substituting  in  one  of  the  preceding  eciuations  (229),  choose  the 
simplest  form.  Complex  equations,  like  Ex.  2,  must  be  simplified 
before  proceeding  with  the  elimination. 

3.    Given  (1)  3  x  +  4  //  =  10,  and  4  x -\-  //  =  <). 
SuooESTiON.  —  Find  the  value  of  ?/  in  (2)  and  substitute  it  in  (1). 

4  j(l)  X  +  2//=:13 

•    }  (2)   2a:  +  3//  =  21. 

5  (1)   40^-3// =  26 
"     ~      -4,/  =  U. 

//  =  10 


6.  J 


(2)   3x 

[(1)0.- 


(2) 


5^3 


10. 


7. 


8. 


(1)    I 


••^  =  1 


(2)    ox  —  3  //  =  10. 
i  (1)   Si)x-\-17  t/=H6 
I  (2)   50  .r-  13// =  17. 


(2) 


1,1 

4^  +  r,//=«- 


10.  < 


0)  r^^  +  .y  =  i<5 


11. 


(2)  ^  +  l=U. 

I  '* 

1(2)  ^  =  ^-1. 

I  ^  3     r> 


r(i)  3x-7//  =  o 

1(2)    U^P^' 


218    .  TEXT-BOOK    OF    ALGEBRA. 

13.    Given  {  1)  ax  -\-  by  =  c  and  (2)  mx  -\-  ny  =p. 
(IJ  ax  =  c  —  by 
c  —  by 


(1.)    ^- 


a 


(2)  m  r-~^\  +  ny^p  (328) 

mc  —  mby  -h  any  =  aj)  (Ax.  3) 

(an  —  bm)  y  =  ap  —  cm  (97,  Ax.  2) 

ap  —  cm        . 

an  —  bm 

The  value  of  x  may  be  found  by  substituting  this  value  of  ?/,  but  in 
literal  exercises  it  is  often  simpler  to  go  back  and  eliminate  the  other 
unknown. 


(I3)  y '- 

c 

—  a2 

b 

(2)  mx 

+  > 

•{*- 

—  ax\ 

wl 

lience  x 

p  (228) 

^^E^Z^.    Ans.  (219,2) 

6m  —  an 

Or,  substituting  the  value  of  y  in  (1)  we  have 

ax  +  b  f^^P-^^A  =  c  (229) 

\an—  bm  I 

whence  x  =  ^^  ~  ^^^  as  before.       (219,  2) 
bm  —  an 

a.  Wlien  equations  are  symmetrical  (i.e.,  such  that  when  corre- 
sponding letters  are  inter  changed^  the  equation  is  not  altered)  one 
answer  may  be  obtained  from  the  other  by  inspection.  Thus,  to  get 
the  value  of  x  from  y,  change  y  to  x,  a  to  ft,  b  to  a,  m  to  n,  and  n  to 
?yi,  which,  can  be  quickly  done. 

14.  (1)    cx-\-dij  =  1  and  (2)    ex  ^  fy  =  1. 

15.  (1)    ax  =  by  and  (2)    hx  -\-  ay  =  c. 


II.— ELIMINATION    BY    COMPARISON. 

231.  Elimination  by  Comparison  is  performed  by  solving 
for  the  same  unknown  in  both  equations,  and  setting  the 
two  values  thus  obtained  equal  to  each  other  (Ax.  7). 


SIMPLE   EQUATIONS. 


219 


232.    Exercise  in  Elimination  by  Comparison. 


7       +       5 


11 


(1,)  r,-ir)r  +  21  7/-7  =  70  (2i)  3a;4-//-hlly  =  99 

(Ax.  .S) 
(lo)   -  15a!  =  72  -  21  //  (22)  3ar  =  99  -  12y 

lo  5 

(Ax.  4) 

7?/ -24 


(2.) 


as-    4y 

7y- 24  =  165 -20// 
27y  =  189 
y  =  7     A71S. 
a;=  3.^-4x7  =  5.     J;?.s. 


(Ax.  7) 


,  (1)    2x  +  7y  =  65 

•    •  (2)    6a:-2y  =  34.       '    I  x 

^  I  (2)    3  -  '/  =  -  1. 


(  (I)    .Sx +.-,,/ =  29     g    J 

(2)  ;•.  X  -  r.  y  =  19. 


6. 


3  a-       // 

('>)    —  —  '    =  2 


(I) 

8- 

x+3 
4      ~ 

7  — 

3 a- -2?/ 
5 

(2) 

42- 

8-y 
3      ~ 

=  24A 

2ar-|-l 
0 

8. 


(1)  21// +  20  a-  =  105 

(2)  77y  — 3()r  =  295. 

2      _      3 

(2)   5(x+3)=3(y-2)+2. 


220  TEXT-BOOK    OF   ALGEBRA. 

(1)    lhx  =  UtJ-\-4j%. 


^'    '   (2)    Ux  =  ^y-21^,. 


10. 


(1)  x-2?j  =  a. 

(2)  2x^H?/  =  0. 


X       y                 ,,..      X          If        2 

(1,)   x  =  a  +  a\                (2J  x  =  2«  +  || 

(Ax.  7) 

26-^ 

(Ax.  2) 

y  =  2}).     Ans. 

(Ax.  h) 

(Ij)  a;  =^  a  +  — r—  =  :J «.     ^ns. 

(229) 

12.    (1)  ?  +  -^  =  TO         (2)    ?  +  '^  =  ». 

^       a       0                     '     cd 

III. -ELIMINATION  BY  ADDITION  AND  SUBTRACTION. 

233.  Elimination  by  Addition  or  Subtraction  is  performed  by 
making  the  coefficients  of  one  of  the  unknowns  the  same  in 
both  equations,  and  then  adding  or  subtracting  the  equations 
member  by  member  according  as  these  coefficients  have  un- 
like or  like  signs. 

1.  In  order  to  make  coefficients  the  same  in  both  equa- 
tions, proceed  as  follows  :  — 

(1.  Keduce  each  equation  to  the  form  ax  •\-hy  ^=  c,  i.e. 
collect  into  the  first  term  all  the  terms  containing  x,  into  the 
second  term  all  the  terms  containing  y,  and  into  the  right 
member  all  the  known  terms. 

(2.  Find  the  1.  c.  m.  of  the  two  coefficients  of  the  un- 
known to  be  eliminated. 

(3.  Multiply  both  members  of  each  equation  by  the 
quotient  obtained  from  dividing  the  1.  c.  m.  by  the  coeffi- 
cient in  that  equation  of  the  unknown  to  be  eliminated. 


SIMPLE    EglATloNS.  221 

2.  Upon  adding  or  subtracting  the  new  equations  mem- 
ber by  meml>er,  the  terms  with  equal  coefficients  cancel  and 
one  unknown  is  thus  eliminated. 

234.    Exercise  in  Elimination  by  Addition  and  Subtraction. 

1.  Given   (1)      ^i:^  +  3  =  i^ 

^^  4  2^3 

(1,)   Ioj:— l>r)//-|-30  =  4.c +  2// 
(U)  11 «  -  27  //  =  -  30  (233.  1,  (1) 

(2.)   9(>-.3x  +  6z/  =  Ca^  +  4y 
(2,)   -<)a:  +  2v/  =  -9()  (233,1,(1) 

(I3)  99  a;  -  243  y  =  -    270  (233,  1,  (3) 

(2,)   -99a;H-22y  =  — 1056  (Ax.  1) 

-221y  =  -1326 
?/  =  6.     .i;i.*f. 
(lij    _  \)x  -I-  i>  X  (;  =  -  90  .-.  X  =  12.     .l;/.s-.      (229) 

2.  (1)  8  a:  -  21  //  =  33  (2)    6  x  -f  35  y  =  177 
(li)   2Ax  -    03 y=    99 

(2,)  24  X  +  140  y  =  708 
203  y  =  609 
y  =  3    Ans. 
(1)    8  x  -  21  X  3  =  33  .-.  X  =  12.     Ans.  (229) 

3.  (1)    10x4-17// =  500       (2)   17x  — 3y=110. 
(li)  48x-f  51y  =  1500 

(2i)  289  X  -  51  y  =  1870 

3;f7x  =3370  (Ax.  1) 

X  =  10.     Ans. 
y  =  20.     Ans,  (229) 


222 


TEXT-BOOK    OF   ALGEBRA. 


4. 


7. 


8. 


10. 


11. 


12. 


13. 


( (1) 

;^x  +  2,/  = 

=  19 

U2) 

2x-3i/  = 

=  4. 

(1) 

< 

M=« 

(2) 

M=«- 

(1) 
(2) 

5 

';'=3 

x-y       X 

2      "^ 

-J-^=«- 

(1) 

2x       y_ 
3    ^4  " 

16 

(2) 

x^l  =  U. 

|(1) 

2           1 
5^-12^ 

=  3 

(2) 

4a:  —  ?/  = 

20. 

(1) 

f  +  l  =  ^^ 

r 

(2) 

.  +  1  =  4. 

!• 

^■(1) 
1(2) 

4^_2j/^ 
5         5 
6  ;r  =  9  y. 

=  4 

(1) 

x  +  37/  _ 

2 

7i 

< 

(2) 

4^  +  5y 
4 

=  8. 

(1) 

3         2 

'lO^ 

(2) 

8^  +  6^  = 

=  7. 

((1) 

4ic  -fSy  = 

=  2.4 

1(2) 

10.2  05 -6 

^  =  3.48, 

SiMl'LK    tAii  A  1  iuN.> 


23 


14      Ul)  ax-\-b,j=p 

((2)  ax^dy=^q. 

((1)  ma:-h«//  =  0 

i(2)  //  =  «a-+/>. 


15 


16. 


17. 


18. 


fa  b 

J  (1)   iT^j  =  3M^ 
l  (2)   rtJ-  +  2  />//  =  ^/. 


(1) 

x  +  i(3 

x  - 

(2) 

§(4  0.  + 

'^l/) 

(1) 

11 
15 

(2) 

5  +  1  = 

x       y 

9 
5 

■)=l+f(y-l) 


Solution. — Inequations   containing   reciprocals   it  is   usually 
best  not  to  clear  of  fractions^  but  to  solve  directly  for  ^  and  * . 


19. 


20. 


0       11 


(i) 

X  "^  y  ~   5 

(2) 

3      4      9 
x^  y~  6 

2       2 

y"5 

10  =  2y 

y  =  5    ^n«. 

(10 

1       11        2       1 
x~15        5"~3*- 

(1) 

5  +  ''  =  .s 

(-0 

15  +  5  =  4. 

(I) 

2li+  12^  =5 

0?                    // 

(2) 

1  _  1  _   1 
//       X       42* 

x  =  3 


(233,  1) 


(Ax.  2) 
(Ax.  3) 

(229) 


224  TEXT-BOOK   OF   ALGEBKA. 


21. 


22. 


23. 


24. 


(1) 

4-! 

a; 

+  27-1  = 
2/ 

42 

(2) 

14 

1_15   U 

a-              y 

=  1. 

f(l) 
1(2) 

4  a?- 

'  +  3y-'  = 
'  +  82/-'  = 

f(l) 

1 

-\'/ 

—  a-  =  4  icy 

~    (-') 

4_ 

!=»■ 

3 

x^ 

5        8 
-y-15 

I  (2)   .),;_22.<-=-^. 


IV. -SPECIAL    METHODS  OF   ELIMINATION. 

235.  Special  Methods  of  Elimination.  In  these  the  elim- 
ination is  attained  by  one  or  other  of  the  three  processes 
already  explained.  The  difference  lies  in  the  preparation 
of  the  equations  for  the  elimination. 

A,    Elimination  by  Undetermined  Multipliers. 

236.  The  General  Process  of  Elimination  by  Undetermined 
Multipliers  for  equations  of  any  degrees  is  called  Bezout's 
Method. 

1.    Given  {1)  hx-ly  =  2Q        (2)  9  ic  -  11 2/  =  44. 

(1 J  5  mx  —  7  my  =  20  ni  (Ax.  3) 

(2)  9x-ny  =  U 

(3  )  (5  m  —  9)  X  —  (1  m  —  ll)y  =  20  m  —  44  (Ax.  2) 

Now,  if  5  m  —9  =  0,  then  x  will  be  eliminated  from  the  equation. 
If  5  m  —  9  =  0,  m  =  ~.     Substituting  this  value  in  (3),  we  have 


.-( 


9  \  9 

7  X  V  -  11 )  2/  =  20  X  -  -  44 


r.  8  40 

Or,  —^„z=z-  _ 


o 

7/  =  .") 

.r  ^  11.  (229) 


SIMPLi:   I'Xa'ATlONS.  225 

If  7  </'  —  11  be  put  «'(iual  to  uoiight,  //  will  In-  eliminated. 

11 
I'laciiig  7  m  —  11  =  «'.  Ill  -    _  .     Substituting  this  vahn'  in  (;J) 

(o  X  —  -  IH   .r  -  0  ^  -JO  X  --  -  44 


8     _ 
7 

88 
7 

X  = 

11 

'•  tf  = 

'). 

(220) 

XoTK.  —  We  readily  see  that  this  iiu*thotl  iuiiounts  to  the  nmlti- 
pUc-ation  of  one  of  the  equations  by  some  number  integral  or  frac- 
tional  whieh  makes  the  coetticients  of  one  of  the  unknowns  the  same 
in  both,  the  multii)lers  being  the  values  of  in.     (233,  1.) 

2.  Given  o  jc  —  5  //  =  45  aiul  18  u-  —  24  //  =  ti24. 

3.  Given  ox  —  7  //  =  20  and  9  a*  —  11  y  =  70. 

4.  (iiven  o  -h  •:  =  <>'^  'Hi'l  .;  +  -^  =  '39.2. 

5.  (liven  X  -[•  aij  ^  b  and  ax  -)-  Z»//  =  1. 


B     Elimination  by  Means  of  one  or  more  Derived  Equations. 

237.  When  One  of  the  Regular  Processes  of  Elimination 
gives  ri.se  to  large  products,  the  work  can  often  be  materi- 
ally shortened  by  means  of  derived  equations. 

If  either  or  both  of  the  given  equations  be  multiplied  by 
some  numl)er  and  the  resulting  equations  added  or  sul>- 
tracted  member  by  member,  we  get  a  new  equation  whicli 
will  he  satisfied  by  the  same  values  of  x  and  y  as  the  given 
equations  (Ax.  a.)  This  new  equation,  it  is  true,  has  both 
X  and  y  still  in  it,  but  with  smaller  coefficients.  Now,  this 
equation  used  with  either  one  of  the  given  equations,  or 
with  another  similarly  derived,  since  two  equations  suffice, 
will  give  the  values  of  x  and  y  by  ordinary  methods  of 
elimination. 


x  + 

y  = 

16 

9x  +  13?/  = 

184 

9x  + 

9y  = 

144 

226  TEXT-BOOK  OF  ALGEBRA. 

1.  (1  )     9  cc  +  13  ?/  =  184     (2)  13  £c  +  19  ?/  =  268 
(li)  27  X  +  39  ?/  =  552 
(2i)  26  X  +  38  y  =  536 
(3 )  ~^+       y=    16  (Ax.  2) 

(1) 

(3i)  

4  */  =  40  (Ax.  2) 

2/ =10 

.•.a-=    6.    .  (229) 

2.  (1)  14  X  -  17  ?/  =  159     (2)  29  x  -  37  //  =  324. 
Suggestion.  —  Multiply  (1)  by  2  and  subtract  from  (2). 

3.  (1)  139  X  + 152  ?/  =  1377     (2)  35  x  +  37  ?/  =  348. 

4.  (1)  755^-564  ?/  =  2074    (2)  1133  x  -  847?/  =  3113. 

5.  (1)  x  +  50?/  =  557  (2)  50x  +  ?/  =  361. 

Sr(;(;i-:sTiON.  —  Add  the  two  equations  and  divide  through  by  51 ; 
then  subtract  the  one  from  the  other  and  divide  through  by  49. 

6.  (I)  59x  +  73y  =  390| 

7.  (1)  23x  — 41?/=  — 

8.  (1)  28x  — 35?/=56 

9.  (1)   {a-\-2h)x—  (a- 
(2)  (a  +  30i/-(a- 

C.    Elimination  by  Method  of  finding  H.  C.  F. 
238.    This  Method  depends  upon  the  Same  Principles  as  the 
last  and  is  really  only  a  special  way  of  applying  them. 

1.     Given  (1)  .5  ^- -  7  ?/ =  -  8  (2)  .5  y  +  82  =  7  u- 

(1,)  5  .r.  -  7  ?/  +  8  =  0  (2j)  7  x  -  5  ?/  -  32  =  0 

7  X  -  .5  ?/  -  82)  5  :/•  -  7  ?/  +  8 
5  X  —  7  ?y  +  8  (J_ 

2)2y/+2?/  — 40 

X  +     ?/  -  20)  .->  .r  -  7  ?/  +  8     (5 
r>  X  +  5  ?/  -  100 
-  12  ?/  +  108  =  0 

-  12  ?/  =  -  108 
7/  =  9 

.-.  x  =  ll.  (229) 


•f         (2) 

73x- 

■f59?/  = 

=  379}. 

2^     (2) 

39x 

-25y  = 

=  63. 

(2) 

29  a:- 

-13y  = 

=  151. 

-2b}!,= 

=  C«c 

-  3  c)  X  == 

.  4  ab. 

simpm:  i:(^uation's.  liiii 

Explanation.  —  Kach  etiuation  is  so  transposoii  that  one  nu'iiibcr 
is  zero.  Then  the  divisor  is  multiplied  through  by  some  factor,  and 
the  result  subtracted  from  the  other  given  equation,  giving  a  new 
equation,  which  is  used  with  the  divisor  (or  dividend),  and  so  on. 
The  last  remainder  does  not  contain  j;  i.e.,  it  has  been  elhnhiated. 
This  remainder,  e(|ual  to  zero,  gives  the  value  of  y. 

2.  (1)    Sx  -  21  t/  =  33  (2)    (>  J-  +  35 //  =  177. 

3.  (1)  l(j  x  +  \7  !/  =  500  (2)  17  .r  -  3  //  =  110. 

239.  Exercise  in  Elimination.  The  choice  of  the  method  is 
left  to  the  student.  Addition  and  Subti-action  usually  gives 
a  more  elegant  form  to  the  solution  than  the  other  two 
methods. 

1.  4  or  -h  9  //  =  51  and  8  a-  —  13  //  =  9. 

2.  7  .r  -  9  //  =  7  and  3  x  +  10  //  =  100. 

3.  4  ./•  -f  S  y  =  2.4  an. I  1 0.2  ./•  —  (*)  //  =  3.4S. 

4.  X  -\-  If  =  24  and  x  =  5  y. 

5.  2  //  -\-  79  =^iyx  and  3  ./■  —  7  =  4  -f  ./•  +  //. 

6.  :£  +  -•"  =  (•,  ami  ^^+'^  =  0. 
3        .)  3         i> 

7.  ^+^  =  :^  +  2and^4--^  =  -^  +  4. 
^       ^       2  4^3        10 

4  5 

8. = and  2  ./•  4-  5  //  =  35. 

5  +  y       12+ J-  ^     •' 

9.  10  J-  =  2  -f  2  //  and  4  //  ^  20  -  1  r. 

10.  •'*  t  ^'  +  '"^  V  -  •■'!  ;"'<'  -^  '  ''  +  10. *•  =  ly- 

•  >  1 

11.  ox  -\-  />//  =  r  —  (/  and  mx  =  n//. 

12.  jj^±J;  =  '!amlS.,-4  =  !»y. 

13.  '+-=l'aM,liLL±iii^  =  /,. 
./•  —  //  X  -{-  a 

14.  .-)  ./•  -h  7  //  :  3  r  +  1 1  : :  13  :  7  and  1 1  r  +  27  :  7  ./'  -|- 

oy::19:ll. 


228  TEXT-BOOK    OF    ALGEBRA. 


16.   2  (2  X  +  ;^  I/)  =  3  (2  ./•  —  o  y)  +  10  and  4  a;  -  3  //  = 
4  (()  y  -  2  .r)  +  8. 

17. +  -^^^ =  2  and '-^  -[-  y/  =  1). 

7  5  11 

18.  l(x-\-  2)  +  ^(y  -  a-)  =  --^  -  ^  ^11^^^  ^  (-  //  -  ^^)  + 
/  4  o 

1  (8  ^  +  0  //  —  4)  =  3  X  +  4. 
o 

19.  2  a; :  y  : :  29  :  4  and  y-^lx^^S  ^  4 //-^  +  13  .r//  -  12.x--^ 

4  y  -  3  ./•  -  1 

on    ^       3       1       18        ,2       1       1  /I    ,    1\    ,    1 

20.  -    I   -  _|_  _  and-  _-  =  ^_  +  -     + 

ic  '  y  '  4      2/        ^     2/      ^  \^     y J      J-^ 

(See  Ex.  18  in  234.) 

21.  _^-  +  _J^  =  2«.  and^-:^l  =  l. 
a  -\-  b       a  —  h  4:  ah 

22.  §^i±iidy^  -  ^-+^-V  =  5  +  -^^^  and 

10  15  5 

9?/  +  5a7  —  8       X  -\-  y       1  X  -\-Q> 
I2  4'^  ^  ~Tl~" ' 

23.  3  «x  —  2hy  ^  c  and  c/'-^ic  +  ^V  =  ^  ^^'• 

10  9 

2a; +  3y- 29  _7a;  — 8  y,^ 
2  3         ~^ 

Suggestion.  —  Put  2  ic  +  3  ?/  —  29  =  ?t  and   1  x  —  8y  +24  =  tj; 
then  the  two  equations  become 

i5_^^8and!!  =  ^, 
u       V  2      3 

from  which  (228)  u  =  2,  and  u  ==  3. 
Substituting  these  values,  we  now  have 

2  ic  +  3  ?/  -  29  =  2,  and  7  X  -  8  ?/  +  24  =  3, 
which,  being  solved,  give  the  values  of  x  and  i/. 


SIMPLE   EQUATIONS.  '2'2y 

+  10//=  10  J- -fssi. 

SUGOEHTiox.  —  Put  =  M   and   5  X  —  8  w  +  44  =  r, 

transposing  lU  ij. 

240.    Problems  containing  Two  Unknown  Quantities. 

1.  When  the  greater  of  two  numbers  is  divided  by  the 
less  tlje  quotient  is  4  and  the  remainder  3 ;  and  when  the 
sum  of  the  two  numl)ers  is  increased  by  38  and  the  result 
divided  by  the  greater  of  the  two  numl)ers  the  quotient  is  2 
and  the  remainder  2. 

Ijet  X  equal  the  greater  number 

and  1/  equal  the  less.  (225,  o.) 

Then  (1)  ^^  =  4  and  (2)  """^-^^^^"^  =  2       (222,  :\) 

(l,)r-4//=    :) 
C2,)x-     y  =  36 


3  y  =  .'« 

(Ax,  2) 

?/=ll 

x  =  4- 

(229) 

2.  A  farmer  paid  four  men  and  six  boys  72  dimes  for 
Liboring  one  day,  and  afterwards  at  the  same  rate  he  i;aid 
three  men  and  nine  boys  81  dimes  for  one  day.  What  were 
the  wages  of  each  ? 

3.  Find  two  nunil»<M's  wljose  sum  is  .sr»71  TJ  and  wjjose 
difference  is  57142S. 

4.  If  A*s  money  were  increased  by  $300  he  would  have 
three  times  as  much  as  H,  but  if  IVs  money  were  diminished 
by  $50  he  would  have  half  as  much  as  A.  Find  the  sum 
possessed  by  each. 


230  TEXT-BOOK    OF    ALGEBRA. 

5.  Ill  an  assembly  of  325  people,  a  measure  was  adopted 
by  a  majority  of  35.  How  many  voted  aye  and  how  many 
nay  ? 

6.  The  planet  Venus  and  the  Earth  complete  their  revo- 
lutions round  the  sun  in  different  times.  At  one  time  they 
are  together  on  one  side  of  the  sun,  that  is  in  conjunction  ; 
at  another  time  they  are  in  opposition,  with  the  sun  between 
them.  When  in  opposition  they  are  159900000  miles 
apart,  when  in  conjunction  only  25700000  miles  apart. 
Supposing  both  to  move  in  circular  orbits,  what  are  their 
distances  from  the  sun  ? 

7.  A  said  to  B,  if  \  of  your  money  were  added  to  ^  of 
mine  the  sum  would  be  $6.  B  replied,  if  \  of  yours  were 
added  to  I  of  mine  the  sum  would  be  $5;^.  What  sum  had 
each  ? 

8.  A  and  B  are  in  trade  together  with  different  sums.  If 
^50  be  added  to  A's  money  and  $20  be  taken  from  B's  they 
will  have  the  same  sum ;  but  if  A's  money  were  3  times 
and  B's  5  times  as  great  as  each  really  is,  they  would  to- 
gether have  $2350.     How  much  has  each  ? 

9.  If  the  smaller  of  two  numbers  be  divided  by  the 
greater  the  quotient  is  .21  and  the  remainder  .0057  ;  but  if 
the  greater  be  divided  by  the  smaller  the  quotient  is  4  and 
the  remainder  1.742.     W^hat  are  the  numbers  ? 

10.  When  Mr.  Smith  was  married  his  age  was  *  of  his 
wife's  ;  12  years  afterwards  his  age  was  g  of  his  wife's. 
How  old  were  they  when  married  ? 

11.  Fifty  laborers  were  engaged  to  remove  an  obstruction 
on  a  railroad  :  some  of  them  by  agreement  were  to  receive 
90  cts.  per  day,  and  others  $1.50.  Tlierewas  paid  them 
just  $48.  No  memorandum  haviiii^-  been  made,  it  is  re- 
quired to  find  how  many  worked  at  each  rate  ? 

12.  A  son  asked  his  father  how  old  he  was,  and  \vas 
answered  thus:   if  you  take  away  5  from  my  years  and 


SIMPLE    KQCATIONS.  281 

divide  the  remainder  by  8  the  quotient  will  be  J  ot  your 
age.  Hut  if  you  iidd  2  to  your  age,  multiply  the  sum  by  :\, 
and  then  subtract  6  from  the  product,  you  will  have  the 
number  of  years  of  my  age.  Wiiat  were  the  ages  of  the 
father  and  son  ? 

13.  What  fraction  is  that  whose  numei-ator  being  doubled 
and  denominator  increased  by  7  the  value  becomes  ^  ;  but 
the  denominator  being  doubled  and  the  numerator  increased 
by  2  the  value  becomes  g  ? 

14.  An  ai)ple-woman  bought  a  lot  of  ajiples  at  1  ct.  each 
and  a  lot  of  pears  at  2  cts.  each,  paying  $1.70  for  the 
whole;  11  of  the  apples  and  7  of  the  i>ears  were  bad,  but 
she  sold  the  good  apples  at  2  cts.  each  and  the  good  jx'ars 
at  3  cts.  each,  realizing  $2.00.  How  many  of  each  fruit  did 
she  buy  ? 

16.    If  a  certain  numl^er  be  divided  by  the  sum  of  its  two 

digits,  the  quotient  is  G  and  the  remainder  3;  if  the  digits 

be  interchanged  and  the  resulting  number  be  divided  by  the 

sum  of  the  digits,  the  (piotient  is  4  and  the  remainder  1). 

What  is  the  number  ? 

SoiATioN.  —  In  order  to  represent  numbers  in  the  Arabic  scale  of 

10  by  letter H^  we  must  say, 

Let  X  =  the  number  denoted  by  the  figure  in  unit's  place, 

and  y  =  the  number  denoted  by  the  figure  in  ten's  plaro, 

and  so  on,  usinj;  other  letters  for  hijjher  orders. 

Then,  lOy  4- j  represents  a  number  of  two  figures, 

100  2  +  10//  +  X  represents  a  number  of  three  figures, 

and  so  on. 

Wlien  tlie  order  of  the  digits  is  reversed, 

10  J  -f  y  =  the  number  of  two  places. 

The  equations  of  the  above  problem  are 

lOy  +  x-3  lOx  +  y-9 

(1)         \^^^         =6 and  (2)  -_^^  =  4. 

16.  Find  a  number  which  is  greater  by  2  than  5  times 
the  sum  of  its  digits,  and  if  9  be  added  to  it  the  digits  will 
be  reversed. 


232  TEXT-BOOK   OF   ALGEniJA. 

17.  Find  tliat  number  of  two  digits,  to  wldcli  if  the  num- 
ber found  by  changing  its  digits  be  added,  tlie  sum  is  121 ; 
and  if  the  less  of  tlie  same  two  be  taken  from  the  greater 
the  remainder  is  9. 

18.  In  11  hours  C  walks  12^  miles  less  than  D  walks  in 
12  hours ;  and  in  5  hours  D  walks  3|  miles  less  than  C  does 
in  7  hours.     How  many  miles  does  each  walk  per  hour  ? 

19.  A  man  has  two  measures.  Nine  of  the  first  or  fif- 
teen of  the  second  will  fill  a  certain  vessel.  Using  both 
measures  13  times  in  all,  how  many  times  is  each  used  ? 

20.  A  banker  has  two  kinds  of  money.  It  takes  a  pieces 
of  the  one  or  b  pieces  of  the  other  to  make  a  dollar.  If  c 
pieces  be  given  for  a  dollar,  how  many  of  each  will  be  used  ? 

21.  A  pound  of  tea  and  3  pounds  of  sugar  cost  $1.20. 
But  if  tea  were  to  rise  50%  and  sugar  10%  they  would  cost 
$1.56.     Find  the  price  per  pound  of  eacli. 

22.  A  grocer  knows  neither  the  weight  nor  the  first  cost 
of  a  box  of  tea  he  had  purchased.  He  only  recollects  that 
if  he  had  sold  the  whole  at  30  cts.  per  pound  he  would  have 
gained  $1.  But  if  he  had  sold  it  at  22  cts.  per  pound  he 
would  have  lost  $3.  Eequired  the  number  of  pounds  in 
the  box  and  the  first  cost  per  pound. 

23.  Two  persons  27  miles  apart  setting  out  at  the  same 
time  are  together  in  9  hours  if  they  walk  in  the  same  direc- 
tion; but  if  they  walk  in  opposite  directions  towards  each 
other,  in  3  hours.     Find  their  rates. 

24.  Find  two  numbers  in  the  ratio  of  5:7  to  which  two 
other  required  numbers  in  the  ratio  3 :  5  being  respectively 
added,  the  sums  shall  be  in  the  ratio  of  9  :  13,  and  the  differ- 
ence of  whose  sums  equals  16. 

25.  Two  trains  set  out  at  the  same  moment,  the  one  to 
go  from  Boston  to  Springfield,  the  other  from  Si)ringfield 
to  Boston.     The  distance  between  the  two  cities'  is  98  miles. 


SIMl'I.i:    KC^TATIONS.  !28o 

They  meet  eacli  other  at  the  end  of  one  honr  and  24  min- 
utes, and  the  train  from  lioston  travtds  as  far  in  4  liours  as 
the  other  does  in  .'1     \\  hat  was  the  speed  of  each  train  '/ 

26.  If  the  sides  of  a  rectangular  liehl  were  each  increased 
by  2  yards,  the  area  wouhl  be  increased  by  220  square  yards. 
If  the  length  were  increased  and  the  breadth  diminished 
each  by  5  yards,  the  area  would  be  diminished  by  185  square 
yards.     What  is  the  area  ? 

SECTIOX  II. 

Simultaneous  Equations  coxtaining  Three  or  More 
Ux K xowx  Qr a xtiti es. 

241.  When  Three  or  More  Unknown  Quantities  appear  in  as 
Many  Equations  the  process  of  eliminating,  properly  con- 
ducted, will  cause  the  unknowns  to  successively  disai)pear, 
and  ultimately  lead  to  the  soluticm  of  the  problem. 

a.  When  there  are  three  equations  containing  three  unknown 
quantities,  we  first  select  that  one  of  the  unknowns  which  can  be 
most  easily  made  to  disappear,  and  proceed  to  eliminate  it  by  usinf, 
say,  the  first  and  second  e(|uations,  obtaining  a  new  (fourth)  equa- 
tion ;  then  tlu»  same  unknown  is  eliminated  by  using  the  first  and 
third  (or,  if  preferable,  the  second  and  third)  equations,  giving  an- 
other new  (fifth)  equation.  These  two  equations,  the  fourth  and 
fifth,  contain  but  two  unknowns,  and  can  be  solved  as  in  Section  I. 
of  this  chapter.  The  wliole  process  becomes  more  intelligible  by 
studying  an  example. 

242.  Examples  of  Equations  containing  Three  or  More  Un- 
known Quantities. 

1.    (Ij       :r-2i/+:]z  =  ()       ] 

(2)  2x-f  .3//-4«  =  20     I 

(3)  3a--2//  +  r>5:  =  26    J 

On  inspection  we  j>erceive  that  of  the  three  unknowns  x 
can  most  readily  be  eliminated,  having  the  smallest  co- 
efficients. 


234 


TEXT-BOOK   OF   ALGEBRA. 


(li)  2x-4.y-\-  6z  =  12 
(2)  2x  -\-3i/  -  4.z==20 
(4)  '  7  //  -  10  ;^  -=    8 

(lo)  3x  ~(j?/  -\-  9.^  =  18 
(3  )  3x  -  2?j  +  5  z  =  26 
(5  )  4y/-    4^^"T 

(5,)  y/-       ^=2 

(52)  7  ?/  -    7z  =  U 

(4  )  7  y  -  10  ^  =    8 

3  ;v  =    6 
^  =    2.  Jws. 
(5i)  ij  -  2  =  2  .:  y  =  4.  Atis. 

(1)  X  -2X4  +  3X2  =  6.-.  it- 


(Ax.  3) 
(Ax.  2) 


(Ax.  2) 
(Ax.  4) 
(Ax.  3) 

(Ax.  2) 

(229) 

=  8.J7^s.(229) 

Note.  —  In  deriving  equations  (4)  and  (5)  aiiij  f/ro  j)airs 
of  the  three  equations  may  be  used. 

2.    (1)      ^  +  2y/-3y^.+     z  =    4.      ^ 

(2)  2  .?•  -     y  +  2  7t  -  3 ,~  =    1 

(3)  5.:^  -3.//-     7^-2,- =  11 

(4)  ;j  ic  -p  4  //  -  r>  y^^  +  (>-  =  _  9  . 
(li)  3  X  +  G  !i  -  9  n  +  3  -  =  12 
(2  )  2  £c  -     y  -f  2  v^  -  3  ;^  =    1 


(5)  hx^hy-lu             =13' 

(I2)  2a;  +  4y/-6«+2;>;=    8 
(3)  5x-37/-     1^-2;^  =  11 

(Ax.  1  1 

(6)  7ir  +     y  -1  n             =19' 
(2i)  4a:-27/  +  4y^,  -G.^=    2 
(4  )  3  :r  +  4  ?/  -  5  ?^  +  C)  ,t  =  -  9 

(Ax.  1) 

(7)  7.r  +  2  7/-7^                =  -7'. 

(Ax.  1) 

Equations  (5),  (G),  and  (7)  constitute  a  new 


set. 


(5  )     5  a?  +  5  ?/  - 
(00  35a.  +  5  v/- 


r)5  y/ 


13 
95 


(8  )  30  X 
(81)  15  a; 


-  28  u  =  82 
-Uu-=  41 


(Ax.  2) 
(Ax.  4) 


SI.MI'LK    KQrATlONS.  235 

62)  14a;  +2y-  Un  =  38 
(7)     7a;  +  2y~       ».  ==  -  7 
(9  )     7  x  -  13  M  =  45    ■  (Ax.  2) 

Equations  (8)  and  (9)  constitute  the  third  set. 

(  9  )     7  j;  —  13  ?«  =  45 

(  8i)  15  a; -14  7/ =41 

(10  )      8 ar  -       u  =  -4  (Ax.  L>.  237) 

(lOi)  104ar-13w=  -  r>2. 

(  9  )       7  a;  —  13  ^  =  45 

97  a;  =  -  97  (Ax.  2) 

.T  =  —    1.     Ans. 

(10)  H  X  -  1  -  u  =  -  4  .'.  >f  =  -  4.     Ans.     (229) 
(6)  7X  -  1  +//- 7  X  -4  =  11) 

.    //  =  —  2     .i/w.     (229) 
(  1)  -  1  +  2  X  -  2  -  3  X  -  ^  +5;  =  4 

.•.;s  =  -3.     yi«..s-.     (229) 

243.  Rule.'  — The  Normal  Process  of  Elimination,  wliere 
tliere  arc  three  or  more  unknowns,  may  be  described  as 
follows :  — 

1.  Select  the  unknown  to  be  first  eliminated.  Then  com- 
bine the  equations  in  pairs  in  the  most  desirable  manner  to 
each  time  eliminate  this  unknown.  However,  the  derived 
equations  must  be  hidependent,  i.e.,  such  that  no  one  of 
them  can  be  derived  from  one  or  more  of  the  others.  (Cf. 
246,  a.)  The  number  of  new  e(iuations  is  one  less  than  the 
given  ecjuations. 

2.  Proc(»e(l  in  the  same  manner  with  the  derived  equa- 
tions to  eliminate  one  of  the  remaining  unknowns.  Then, 
in  like  manner,  with  the  set  thus  obtained,  and  so  on.  until 
the  value  of  the  hist  remaining  unknown  is  found. 

3.  Substitute  the  value  of  the  last  unknown  in  one  of  the 
l)receding  eijuations  contiiining  but  two  unknown  ([uantities 

>  A  far  more  abridged  anil  elegant  method  of  solving  these  problems  Is  by 
means  of  Determinants. 


236  TEXT-BOOK   OF    ALGEBRA. 

and  the  value  of  the  other  is  immediately  known.  Substi- 
tute these  values  in  an  equation  which  contains  a  third  un- 
known, and  so  on. 

a.  If  in  any  case  one  or  more  of  the  equations  do  not  contain  one 
of  the  unknowns,  it  will  usually  be  better  to  regard  them  as  belong- 
ing to  the  second  set. 

b.  Special  metho^te  of  elimination  in  particular  problems  fre- 
quently far  surpass  the  normal  process  in  elegance  and  brevity. 

244.  Exercise  in  Equations  containing  Three  or  More  Un- 
known Quantities. 

[(1)  :r+y  +  ^  =  6 

1.  -j  (2)  5  ic  +  4  ?/  +  3  ,^  =  22 

[  (3)  15  X  -h  10  ?/  +  6  ^  =  53. 
[(1)  x^2y  =  2'S 

2.  <^  (2)  3  :^  +  4  ^  ==  57 

[(3)  57/  +  C^  =  94. 

Suggestion.  — Eliminate  z  between  (2)  and  (3),  and  com- 
bine the  new  equation  with  (1). 

r(l)  a:  +  .y-.^  =  132 

3.  -^(2)  x-y-^z  =  (\5A 

[(3)   -x^y^x=-\.2. 

(1)  ^+J/_^2,v  =  21 
3 

(2)?/_±-_3x=-65    . 
(3)  ■S.r  +  y-.-_3g_ 

f  (1)   i  {.^  +  ~--  '-.)  =  //  -  .t 
5.        (2)  =2.»--ll 

1(3)  =!)-(.r  +  2.t) 

f  (1)  and  (2)  y +  ^_=  •1±5  =  ^_.±J/ 

&■  \  '    ^         3  2 

1(3)  a^  +  y  +  .t  =  27. 


siMi'Li:  i:(,M-.\  rioNs. 


237 


8. 


9. 


10. 


11. 


12. 


(1),  (2)  .r: //:..  =  5:1-:  13 

(i.e.,  .r  :  //  ::  5  :  12  and  jc  :  z  ::  ii :  lo) 

(IV)  ,-  +  //  -f  -  =  27. 

-.  (2)  i  (.r  +;•)  =  <>-// 

[(•^)  1  (.'•-::)  =  2// -7. 

[(1)  .i--h<<  ^  !/  +  .-: 

^  (2)  //  +  ''=  2./- +  2;. 

[(;i)  x  +  a  =  :\j'-^:\^. 

[(1)  «.r  +  /y//  =  r 
^  (2)  rj-  -h  <r:;  =  A 
[  (.S)  A.i;  -h  t-y  =  .^ 

a;      y      s 


_  *i  _  i  =  _  i(».4 

f  1!!  -  «  =  14.!.. 


X 

(2)  _1 


oy 


1         2 
.:J  ic       2  y       ^' 


=  m 


o  a;       2  y       ;i' 

Suggestion.  — Examples  11  and   12  are  to  be  solved  as  recipro- 
cals.    Clear  12  of  numerical  denominators.     (See  Ex.  l8,  234.) 

13.  (1)  xy^yz^zx=  i)xyz;    (2)  yz  ■}- 2 zx  -  3xy  = 
—  4  xyz ;  (3)  S  yz  —  2  zx  -\-  xy  =:  4:  xyz. 

Suggestion.  —  Divide  each  of  these  equations  through  by  xyz. 

14.  (1)  X  +  y  =  1-'?  (2)  !/  -  ^  =  3;  (3)  .^  +  «  =  7 ;  (4) 
u-\-x  =  8. 

Suggestion.  —  First  add  the  equations,  and  divide  by  2. 

15.  (l)x-hy  =  16;  (2).^+x  =  22;   (3)y  +  .v=28. 


238  TEXT-BOOK   OF   ALGEBRA. 

245.    Problems  Involving  Three  or  More  Unknown  Quantities. 

1.  Determine  tliree  numbers  such  that  their  sum  is  9; 
the  sum  of  the  first,  twice  tlie  second,  and  three  times  the 
third,  22 ;  and  the  sum  of  the  first,  four  times  tlie  second, 
and  nine  times  the  third,  oS. 

2.  A  and  B  together  possess  only  §  as  much  money  as  C ; 
B  and  C  together  have  6  times  as  much  as  A ;  and  B  has 
^680  less  than  Aand  C  together;  how  much  has  each  ? 

3.  A  boy  bought  at  one  time  2  apples  and  5  pears  for 
12  cts. ;  at  another  3  pears  and  4  peaches  for  18  cts. ;  at 
another  4  pears  and  5  oranges  for  28  cts. ;  and  at  another 
5  peaches  and  6  oranges  for  39  cts. ;  required  the  cost  of 
each  kind  of  fruit. 

4.  A  gentleman  divided  a  sum  of  money  among  his  four 
sons,  so  that  the  share  of  the  oldest  was  ^  of  the  sum  of  the 
shares  of  the  other  three,  the  share  of  the  second  ^  of  the 
sum  of  the  other  three,  the  share  of  the  third  \  of  the  sum 
of  the  other  three ;  and  it  was  found  that  the  share  of  the 
eldest  exceeded  that  of  the  youngest  by  $14.  What  was 
the  w^hole  sum,  and  what  Avas  the  share  of  each  son  ? 

5.  A  person  has  two  horses  and  two  saddles,  all  of  which 
are  worth  $26o.  The  poorer  horse  and  better  saddle  are 
worth  $5  less  than  the  better  horse  and  poorer  saddle,  while 
the  better  horse  and  better  saddle  are  worth  $45  more  than 
the  poorer  horse  and  poorer  saddle ;  and  the  horses  are 
w(jrth  5  times  as  much  as  the  saddles.  What  is  the  value 
of  each  horse  and  saddle  ? 

6.  The  average  age  of  three  persons  is  41  years.  The 
average  age  of  the  first  and  sec^ond  is  37  years,  and  of  the 
second  and  third  is  29  years.     Find  their  ages. 

7.  The  average  age  of  A,  B,  and  C  is  a.  The  average  age 
of  A  and  B  is  h,  and  of  B  and  C  is  e.     What  are  their  ages  ? 

8.  Three  numbers  are  in  the  ratio  3:4:5.  If  to  fivefold 
the  first  number  we  add  fourfold  the  second  number  and 


SIMPLK    IJ.M    Al'inNS.  289 

threefold  the  third    mmilu'r.  the   sum  will   he  .'U;").     Name 
tlie  three  numbers. 

9.  If  A  and  B  eau  perform  a  certain  work  in  12  days, 
and  A  and  C  in  15  days,  and  B  and  C  in  20  days,  in  what 
time  could  each  do  it  alone  ? 

10.  A  number  is  expressed  by  three  figures  whose  sum 
is  11.  The  figure  in  the  place  of  units  is  double  that  in  the 
place  of  hundreds,  and  when  297  is  added  to  this  number 
the  sum  obtained  is  expressed  by  the  figures  of  this  num- 
ber reversed.     What  is  the  number  ? 

11.  Roads  join  four  cities,  A,  B,  C,  D,  thus  forming  a 
(quadrangle.  If  I  go  from  A  to  T>  through  B  and  C,  I  nnist 
pay  $6.10  hack  fare.  If  I  go  from  A  to  B  through  D  and 
C,  I  must  pay  $5.50.  Going  from  A  to  C  through  B,  I  pay 
the  same  as  from  A  to  C  through  D.  On  the  other  hand, 
from  B  to  I)  through  A  costs  40  cts.  less  than  from  B  to  J) 
through  ('.  What  are  the  distances  A  B,  B  C,  C  D,  and 
D  A,  if  the  fare  is  10  cts.  i)er  mile  ? 

12.  A,  B,  and  C  in  a  hunting  excursion  killed  90  birds, 
which  they  wish  to  share  equally ;  in  order  to  do  this  A, 
who  has  most,  gives  to  B  and  C  as  many  as  they  already 
have ;  next,  B  gives  to  A  and  C  as  many  as  they  had  after 
the  first  division ;  and  lastly,  C  gives  to  A  and  B  as  many 
as  they  each  had  after  the  second  division.  It  was  then 
found  that  each  had  the  same  nund)er.  How  many  had 
each  at  first  ? 

13.  A  piece  of  work  can  be  completed  by  A,  B,  and  ('  in 
10  days;  by  A  and  B  togetluM-  in  11  days,  su])p<)sing  ('  to 
have  worked  5  days  and  left  ntt  ;  liy  \\  an<]  ('  it  !*>  works  15 
days  and  C  works  .30  days.  liow  long  will  it  take  each 
alone  to  do  the  work  ? 


240  TEXT-BOOK    OF   ALGEBRA. 


CHAPTER   X\  1. 

INDETERMINATE  AND   REDUNDANT   EQUATIONS. — 
PROBLEMS   IN    ALGEBRA. 

SECTION   I. 

Indeterminate  Et^uATioNS. 

246.  Indeterminate  Equations  occur  wlieii  the  number  of 
unknown  quantities  exceeds  the  number  of  Independent 
Equations.     (See  225,  b.) 

a.   Equations  seemingly  different  sometimes  reduce  to  the  same 
equation.     Hence  two  given  simultaneous  equations  might  be  mde- 
terminate.     Thus,  (1)  2  (8  x  -  17i)  =  -  H  y. 
(2)    10x  +  ly  =  m- 
Upon  clearing  of  fractions  and  transposing,  both  become 
60  ic  +  42  ?/  =  350. 

247.  Examples  of  Indeterminate  Equations.  These  may 
be  classified  into  two  kinds:  those  on  which  are  imposed 
no  restrictions  ;  and  those  in  which  only  positive  integral 
values  are  permissible.  The  latter  are  often  called  Dio- 
phantine  Equations. 

a.  An  example  of  the  former  kind  was  given  in  225,  b.  It  is 
there  made  plain  that  any  number  of  results  can  be  obtained  from 
such  an  equation,  every  one  of  which  is  a  solution.  Also  two  or 
more  equations  may  be  reduced  by  elimination  to  a  single  equation 
containing  two  or  more  unknowns,  whose  solution  would  be  like 
that  in  225,  5.  All  that  can  be  done  with  these  equations,  then,  is 
to  assign  arbitrary  values  to  one  or  more  of  the  unknowns,  and  de- 
rive the  corresponding  values  of  the  remaining  one.  These  equa- 
tions find  an  appropriate  use  in  analytic  geometry. 


SIMPLE    EQUATIONS.  241 

b.  If,  however,  the  answers  be  conhned  to  integral  positive  num- 
bers, the  problem  becomes  definite,  and  may  be  solved,  although  the 
method  of  solution  is  very  unlike  that  of  other  equations. 

1.  Given  r}x  -f-  11  y  =  47  ;  required  to  find  the  pairs 
of  i)ositive  integral  values  of  x  and  y  which  satisfy  the 
equation. 

5x  =  47-lly  (Ax.  2) 

x  =  ilz;lij^  =  0-2y  +  ^  (162) 


0 


x-\)  +  'J!/  =  '-y^  (Ax.  2) 

o 

Since  x  and  y  are  to  be  whole  numbers,  the  integral 
quantity,  aj  —  9  -|-  2  y,  must  be  a  whole  number ;  and  if  the 
left  side  of  the  equation  is  a  whole  number,  the  7'lf/ht  side 
must  be  a  whole  number  also.     Consequently  values  of  y 

must  be  selected  which  will  make  — —^  an  integer  (posi- 

o 

tive,  negative,  or  zero). 

Such  values  are  y  =  2,  making 

^7^  =  0  (for  0  H-  T)  =  0) 
5 

y  =  ~.  making  --7  "'^  =  —  1 
5 

y  z=  12,  making '1  s  —  2 

5 

Finding  the  corresponding  values  of  x  by  substitution 
(229),  we  have  y  =  2,  a;  =  5 ;  y  =  7,  x  =  —  6.  But  negative 
values  are  excluded.  Furthermore,  larger  values  of  y  would 
give  still  larger  negative  values  of  x.  Hence  x  =  T),  y  =  2, 
are  the  only  pair  of  integral  roots. 

2.    Given  41  or  -|-  11  y  =  79() 

!/  =  ^-^ =  <  1  -  3  J-  +  — ^— .  (162) 


242  TEXT-BOOK    OF   ALGEBRA. 

As  before  ~~. —  must  be  an  integer,  and  the  value  of  x 

must  be  found  from  it  by  trial.  We  may,  however,  derive 
a  shti2)ler  expression  from  which  to  find  the  value  of  x. 

For,  if  —    —  is  an  integer,  any  whole  number  of  times  it 

must  also  be  an  integer.  Let  us  multiply  this  fraction  by 
7  (since  7  times  8  is  one  greater  than  a  multiple  of  11),  and 
reduce  the  new  improper  fraction  to  a  mixed  quantity. 

Now,  if  5  —  5x-\- is  integral,  since  the  part  5  —  ox 

is  integral,  the  fraction  is  likewise  integral.     From 

this  fraction,  then,  values  of  x  may  be  found  which  must 
make  it  integral. 
Putting 

.  =  8,  (1=^  =  O),  ,  =  ^p  '  =  ''■  ''"'■ 
X  =  19,  (^^  =  -  1^  .y  ^790-41  X19^^       ^,^ 

No  other  values  of  x  will  give  y  positive. 

248.    Rule  for  tiie  Solution  of  Indeterminate  Equations. 

1.  When  there  are  more  than  one  of  the  given  equations, 
reduce  them  by  elimination  to  a  single  equation  containing 
two  or  more  unknown  quantities. 

2.  If  more  than  two  unknowns  remain  in  the  single 
equation  thus  left,  arbitrary  values  must  be  assigned  to  all 
but  two  of  them. 

3.  Solve  for  that  one  of  the  unknowns  most  readily 
found  in  terms  of  the  other,  and  reduce  its  value  (when 


SIMPLE    EQUATIONS.  243 

(livisil)le)  to  the  lorin  ot  a  inixiHl  quantity.  The  iniutjon 
thus  ubtuined  can  Impiently  he  transtoinied  into  a  simpler 
fraction  by  multiplyiuL;  it  1)\  ^oiue  number  (always  less 
than  its  denominator)  wliich  will  make  the  coefficient  of 
the  unknown  one  greater  than  some  multiide  of  the  denom- 
inator, and  then  reducing  this  new  fraction  to  a  mixed 
quantity.  The  fractional  part  of  this  mixed  (|n:intity  may 
be  used  instead  of  the  original  fraction. 

4.  Supply  successively  such  values  to  the  unknown  in 
the  numerator  of  the  fraction  as  will  make  it  zero,  or  some 
multiple  of  the  denominator,  and  each  time  find  the  cor- 
responding value  of  the  other.  Such  pairs  of  values  will  be 
answers  so  long  as  they  are  both  positive. 

Remark.  —  The  method  here  given  is  very  elementary.  Other 
letters  used  as  auxlHaries  in  the  solution  are  often  employed,  but 
their  use  is  not  essential,  and  they  have  not  been  introduced  in  the 
explanations  given  above. 

249.   Exercise  in  the  Solution  of  Indeterminate  Equations. 
1.    2  a;  -}-  3  //  =  L^5.  2.   5  ic  +  7  y  +  4  =  50. 

3.  x4-3y-f-2«=  10. 

Suggestion.  —  Here  it  will  be  necessary  to  assign  arbitrary 
values  to  one  of  the  letters,  say  z.  Zero  values  being  excluded,  z 
may  equal  1,  or  2,  or  3.  These  values  give  three  different  indeter- 
minate equations  to  be  solved  for  x  and  y. 

4.  2;;7:{=  Vix-\-24y. 

5.  17.r +5;5y—  123  =  411  -  19  .r -f-  15  y. 

«■  1S3^^^^;I51.         '  ^«-=^«"'> 

I  (2)  4  a;  —  5  //  —  G  ,t;  =  —  GG. 
J  (1)   29//  =  8a^-4 
■     /  (2)  4o2^17x  —  z. 
\(\)  2x  +  3i/^7z  =  V.) 
9.    i  (2)   5ar  +  8y  +  ll.v-24 

[  (3)  7  X  -h  11  y  -h  -1  -  =  43.  ^^.M.  246,  '') 


244  TEXT-BOOK   OF   ALGEBRA. 

250.    Problems  involving  Diophantine  Equations. 

1.  Separate  71  into  parts  of  which  the  one  is  divisible  by 
5  and  the  other  by  8  without  remainders. 

2.  "  Had  I  two  times  as  many  eggs  as  I  now  have  "  said 
one  peasant  girl  to  another,  "  and  you  seven  times  as  many 
as  you  now  have,  and  I  were  then  to  give  you  one  egg,  we 
should  each  have  the  same  number."  How  many  eggs  had 
each  ? 

3.  In  how  many  ways  may  100  be  divided  into  two  parts 
one  of  which  shall  be  a  multiple  of  7  and  the  other  of  9  ? 

4.  What  is  the  least  number  which  when  divided  by  5 
and  3  leaves  remainders  3  and  2  respectively.  How  many 
such  numbers  are  less  than  100  ? 

Suggestion.  —  Let  z  be  the  nmnber  and  x  and  y  the  quotients. 

5.  A  person  bought  40  animals  consisting  of  pigs,  geese, 
and  chickens  for  f  40.  The  pigs  cost  $5  a  piece,  the  geese 
$1,  and  the  chickens  25  cts.  each.  Find  the  number  he 
bought  of  each. 

6.  Divide  17  into  three  such  parts  that  if  the  first  be 
multiplied  by  5,  the  second  by  4,  and  the  third  by  7,  the 
sum  of  these  products  is  80. 

'  7.  A  number  consisting  of  three  digits,  of  which  the 
middle  one  is  2,  has  its  digits  inverted  by  adding  198. 
What  is  the  number  ? 

8.  A  farmer  buys  oxen,  sheep,  and  hens.  The  whole 
number  bought  is  100  and  the  total  price  $100.  If  the 
oxen  cost  $35,  the  sheep  $3,  and  the  hens  2o  cts.  each,  how 
many  of  each  did  he  buy  ? 

9.  A  boy  sees  that  he  can  buy  oranges  at  2  cts.,  3  cts., 
4  cts.,  5  cts.,  or  6  cts.  apiece  and  spend  all  his  money ;  but 
if  he  buys  at  7  cts.  apiece  he  will  have  5  cts.  remaining. 
How  much  money  has  he  if  he  has  the  least  sum  possible  ? 

Suggestion,  —  The  1,  c,  m,  of  2,  3,  4,  5,  and  6  is  00.  Let  60  x  = 
Ills  money, 


SIMPLK    KQIATIONS.  245 

10.  Find  four  integral  numbers  such  that  the  sum  of  the 
first  three  shall  be  18 ;  the  sum  of  the  first,  second,  and 
fourth  16 ;  and  the  sum  of  the  first,  third  and  fourth  14. 

SECTION   II. 
Redundant  Equations. 

251.  Redundant  Equations  occur  when  the  number  of  equar 
tions  exceeds  the  number  of  Unknown  Quantities.  These 
equations  may  be  chissed  into  two  kinds  : 

1.  Compatible  Redmidant  Ef/uatlojis  are  such  as  are  satis- 
fied by  values  of  the  unknowns  derived  from  the  other 
equations  of  the  set  to  which  they  beh)ng. 

a.  Redundant  equations  were  used  in  237.  The  new  equations 
formed  out  of  the  old  were  satisfied  by  the  same  values  of  the  un- 
knowns,  i.e.,  were  compatible.  When  new  equations  were  con- 
structed an  equal  number  of  old  ones  were  dropped. 

We  can  also  have  independent  redundant  equations.  Thus,  (1) 
7  X  -H  2  7/  =  .>i,  (2)  12  X  +  5  y  =  94,  (3)  14  x  -  11  y  =  70,  are  all  sat- 
isfied by  X  =  7,  y  =  2.  All  that  can  be  done  with  such  equations  is 
to  test  for  comi)atibiIity,  or  incompatibility. 

2.  Incompatible  Redundant  Equations  are  such  as  are  not 
satisfied  by  the  values  of  the  unknowns  obtained  from  the 
other  equations  of  the  set  to  which  they  belong.  Thus,  in 
(l)2a;  +  3y  =  23(2)4x-5y  =  -9(3)  llx  +  17y  =  6, 
(1)  and  (2)  give  r  =  4,  y  =  5 ;  (2)  and  (3)  give  a;  =  —  1, 
//  =  1  ;  (1)  and  (3)  give  x  =  373,  y  =  -  241. 

SECTION  III. 

\koativk  and  Inconsistent  Solutions  in  Pkoblkms  ixvolv- 
ixo  SiMiM.K  Equations. 

252.  Problems  involving  Arithmetical  Inconsistencies. 

1.  If  from  f  of  a  certain  number  1  be  subtracted,  the  re- 
mainder equals  the  sum  of  twice  the  number  divided  by  7, 


246  TEXT-BOOK    OF   ALGEBRA. 

five  times  the  number  divided  by  14  and  3.     What  is  the 
number  ? 

|c.-l  =  ^'^  +  ff  +  3  (219,1) 

a;  =  -  224  (219,  2) 

Here  the  negative  answer  points  to  some  absurdity  in  the 
problem  viewed  as  an  arithmetical  one.     On  examination 

we  find  that  —  -\ is  greater  than  — ;-,  and  should  be  di- 
minished rather  than  increased  to  produce  — . 

2.  A  father's  age  is  40  years  ;  his  son's  age  is  13  ;  in  how 
many  years  will  the  age  of  the  father  be  four  times  that  of 
the  son  ? 

40-f  ic  =  4(13  +  a;)  (219,1) 

X  =  —  4  (219,  2) 

In  this  question  the  negative  answer  shows  that  the  tacit 
assumption  that  the  epoch  named  would  be  in  the  future 
was  wrong.  The  question  should  have  read  "  how  long 
ago,"  and  the  equation  have  been 

40  —  a?  =  4  (13  —  x),  whence  cc  =  +  4. 

3.  A  and  B  went  into  business  agreeing  to  divide  the 
profits  in  a  certain  way.  Twice  B's  money  diminished  by 
$4250  indicated  A's  financial  standing  in  the  company 
when  they  began.  A  made  $4000  and  B  made  $750  when 
it  was  found  that  A  had  $500  more  than  B.  How  much 
had  each  when  they  began  ? 

(1)  X  +  4250  =  2  2/ 

(2)  X  +  4000  =  7/  +  750  +  500  (219,  1) 
a;  =  —  1250  =  A's,  and  y  =  1500  =  B's  (219,  2) 

The  result  indicates  that  A  was  in  debt  when  he  first 
be^jan  business. 


SIMPLE   EQUATIONS.  247 

4.  A  man  worked  7  days  and  had  his  son  with  him  3  days, 
and  received  as  jjay  $2.20.  He  afterwards  worked  5  days 
and  hiKl  his  sou  with  him  one  day,  and  received  for  wages 
$1.80.  What  was  the  father's  daily  wages,  and  what  was 
tlie  effect  of  the  son's  presence  ? 

(1)  7a;  +  3y  =  220 

(2)  5x  +  f/=  ISO 

a;  =  40  cts.  father's  wages,  y  =  20  cts.  son's  expense, 
i.e.,  the  father  paid  out  20  cts.  a  day  for  the  sou. 

5.  A  and  B  travel  in  the  same  direction  at  the  rate  of  6 
mi.  and  4  mi.  respectively  per  hour.  A  arrives  at  a  certain 
place  P  at  a  certain  time,  and  at  tlie  end  of  8  hours  from 
that  time  B  arrives  at  a  certain  place  Q.  Find  when  A  and 
B  meet. 

P                                       Q                              R 
H 1 1 

1.  Su[)pose  the  distance  P  Q  50  miles. 

Let  X  =  the  number  of  hours  from  the  time  A  is  at  P,  till 
they  meet  at  R.  Then  since  A  travels  at  the  rate  of  0  miles 
per  hour,  the  distance  P  R  is  6  x  miles.  Also  B  goes  ov(»r 
the  distance  Q  R  in  x  —  8  hours,  so  that  Q  R  equals  4  (x 
-  8)  miles.     Now  P  R  =  P  Q  +  Q  R.     Hence 

Gx  =  r)()  +  4(a;  — 8) 

x  =  9  (219,  2) 

2.  But  if  the  distance  P  Q  is  20  miles  the  equation 
would  be 

6j-  =  2()  +  4(x-8) 
Wlience  a:  =  —  G. 

The  negative  value  of  x  indicates  that  they  met  to  the 
left  of  P  instead  of  at  the  riglit.  It  is  plain  that  in  8  hours 
B  would  walk  32  miles,  a  number  greater  than  20.  Con- 
sequently they  met  before  A  reached  P. 


248  "  TEXT-BOOK   OF   ALGEBRA. 

3.  N^ext  suppose  A  travels  4  miles,  and  B  G  miles  per 
hour,  and  suppose  P  Q  =  50  again.     Then 

4  X  =  50  +  6  (x  —  8) 

In  8  hours  B  travels  48  miles.  He  would  therefore  be 
just  2  miles  beyond  P  at  the  time  A  arrived  there.  They 
liad  met  one  hour  before  A  arrived  at  P. 

4.  Lastly  suppose  A  and  B  travel  in  as  3,  but  that  P  Q 
is  20  miles. 

4.x  =  20  +  G(:r  — 8) 
.-.  :r  =  14 

Cases  (1  and  (4  are  in  accord  witli  the  idea  of  meeting  in 
the  future  as  implied  in  the  statement,  while  (2  and  (3  con- 
tradict this  supposition. 

6.  A  grocer  has  twO  barrels  of  molasses,  one  of  whicli 
contains  twice  as  much  as  the  other.  From  the  larger  cask 
he  draws  16  gals,  and  from  the  smaller  10  gals.  Then  after 
a  fourth  of  what  remained  in  the  larger  cask  had  been  witli- 
drawn,  the  two  casks  were  found  to  contain  an  equal  num- 
ber of  gallons.     How  much  did  each  cask  hold  ? 

Let  2  X  =  tlie  number  of  gals,  in  the  larger 
and     X  =    '•         ''         "     "      "     "    smaller. 
Then    f  (2  x  —  10)  =  x  —  10 

x  =  4 

2a;  =  8 

Here  the  answers  are  positive,  but  that  does  not  save  us 
from  the  absurdity  of  drawing  16  gals,  from  an  8  gal.  cask, 
and  10  gals,  from  a  4  gal.  cask ! 

Verifying  the  equation,  3  x  —  8  =  —  6.  Thus  we  learn 
that  the  existence  of  the  minus  quantities  in  the  process  of 
solution  vitiates  the  result  for  the  arithmetical  problem. 

7.  Ex.  33,  Art.  224,  illustrates  the  point  brought  out  in 
Ex.  6  still  further.     By  the  statement  of  that  problem  we 


SIMPLK    FX^UATrONS.  '240 

are  led  to  suppose  that  the  amount  invested  at  G  ^  is  only 
a  part  of  the  $12000.  The  solution  shows  that  the  man 
borrowed  $132(K),  on  part  of  which  he  paid  4  ^  and  the 
other  pai-t  r>  '/i .  He  got  6  %  on  the  whole  $25200.  Six 
per  cent  on  $\'J(M)()  amounts  to  only  $720'. 

253.  The  Problems  of  the  preceding  Article  illustrate, 

1.  That  a  iiegative  result  indicates  either  some  arithmeti- 
cal incongruity  in  the  statement  of  the  ])roblem,  or  at  least 
that  the  number  obtained  as  the  answer  should  be  tiiken  in 
a  sense  conti-ary  to  that  implied  in  the  statement  of  the 
l)roblem. 

2.  That  positive  results  do  not  iipr'«>ssarily  ])rove  that 
l)roblems  ai*e  satisfactorily  soK.d.  Ilcuce  all  ])roblenis 
which  do  not  admit  of  algebraic,  interpretation  (i.e.,  tliose 
which  do  not  admit  of  both  positive  and  negative  values  for 
their  quantities),  when  solved  by  algebraic  methods  should 
have  their  answers  tested  for  arithmetical  ccmsistency. 

SKCTION    IV. 
LiTKKAi,  I'lMuu.KM"^.  —  Generalization. 

254.  The  Generalization  of  a  Problem  is  attained  by  repla- 
cing the  known  iuinil)ers  by  lrtiri>  and  then  deriving  its 
solution. 

(I.  Problems  are  ronimonly  first  stiiditd  liy  usin^  particular  num- 
bers; afterwards,  on  account  of  the  freciucncy  witli  wlil<'h  they 
occur,  they  are  generalized.  After  such  a  sohition  has  been  worked 
out,  all  that  is  necessary  afterwards  is  to  substitute  the  numbers  of 
any  particular  case  of  the  problem  in  the  answer  of  the  generalized 
solution,  and  reduce  as  in  81. 

255.  Exercise  in  the  Solution  of  Literal  Problems  and  Gen- 
eralization. 

ii.  In  some  of  the  first  problems,  and  also  those  difficult  to  state, 
reference  to  a  particular  form  of  tli<-  umxcii  jnohlem,  or  one  similar 


250  TEXT-BOOK   OF   ALGEBRA. 

to  it,  will   be  made.     The  student  will   find  these  references  very 
helpful  if  aid  is  needed  in  the  statement. 

6.  The  following  set  of  problems  includes  such  as  contain  some- 
times one,  at  other  times  more  unknown  quantities.  The  student  is 
left  to  decide  to  which  class  any  given  problem  ought  to  be  assigned. 

1.  (1.  Particular  form.  —  Divide  |>183  between  two  men 
so  that  ^  of  what  the  first  receives  shall  equal  ^\  of  what 
the  second  receives. 

i^~x  =  j%  (183  -  x)  (219,  1) 

cc  ==  63 ;    183  —  x  =  120  (219,  2) 

(2.  Generalized  form.  —  Divide  $a  between  two  men 
so  that  -  of  what  the  first  receives  shall  be  equal  to  ^  of 
what  the  second  receives. 

--x  =  L(a-x)  (219,1) 

np  -f-  mq  ^       '     ^ 

(3.  Special  case.  Solution  by  substitution.  —  Divide 
168  into  two  parts  so  that  -§  of  one  shall  equal  {^  of  the 
other. 

Here  a  =  168,  m  =  5,  n  =  9,  p  =  5,  q  =  12.     Hence 

^  =  o"^v!^rc^^r^^io  =  ^^-     ^ns.     168  -  72  =  96.     Ans. 
9  X  5  -f  5  X  12 

Verification.     §  of  72  =  f>^  of  96. 

2.  A  boy  bought  an  equal  number  of  apples,  lemons,  and 
oranges  for  c  cts. ;  for  the  apples  he  gave  I  cts.,  for  the 
lemons  m  cts.,  and  for  the  oranges  n  cts.  apiece.  How  many 
of  each  did  he  purchase  ?     See  Ex.  4,  224. 

3.  A  father  is  now  a  years  of  age,  and  his  daughter  b 
years  of  age.  How  many  years  ago  was  the  father's  age  n 
times  that  of  the  daughter  ?     See  ex.  9,  224. 

Obtain  the  answer  in  the  following  cases  by  substitution. 


SIMl'LE    EQUATIONS.  251 

a  =  51,  />  =  24,  n  =  2\ ;    ti  =  75,  h  =  19,  7i  =  5 ;    a  =  ;]7,  h 
=  12,  Ji  =  ;i. 

4.    The  sum  of  two  miiuUt'is  is  .v,  ami  their  (lirt'crt'iicc  </  ; 
what  are  the  niiiul^ers  ? 


Solution,   Let  x  -|-  //  =  .s- 

«  +  //  =  ••>• 

and     X  —  1/  ^  d 

X  —  t/  =  d 

then  2  ar          =s-\-d 

2  //  =  .s-  —  rf    (Ax.  a) 

S  -f-  f / 

s  —  </ 

//=      .,        (Ax.  4) 

These  formulae  may  be  stated  in  the  form  of  theorems. 

To  find  the  greater  of  the  two  numbers  take  half  the  sum 
of  the  sum  and  difference. 

To  find  the  less  of  the  two  numbers  take  halt  the  differ- 
ence of  the  sum  and  difference. 

Given  s  =  64,  c?  =  2G  ;  s  =  195,  (/  =  14  ;  s  =  300,  d  = 
312. 

5.  Divide  a  into  two  parts  such  that  the  difference  be- 
tween one  part  and  b,  shall  eijual  7i  times  the  difference 
between  the  other  part  and  c.  Put  a  =  A,  b  =  3,  7i  =  2,  c 
=  1. 

6.  Two  men  a  miles  apart  travel  towards  eacdi  other,  one 
m  miles,  the  other  n  miles  an  hour.  In  how  many  hours 
will  they  meet  ?     Put  a  =  49,  7n  =  4^,  w  =  3'^. 

7.  A  farmer  sells  n  horses  and  b  cows  for  c  dollars.  And 
at  the  same  prices  a'  horses,  and  b'  cows  for  c'  dollars.  Wliat 
is  the  price  of  each  ?  Given  in  a  particular  problem  that  a 
=  10,  i  =  9,  c  =  $1600,  a'  =  12,  //  =  7,  c'  =  $1720.  Find 
the  prices. 

8.  A  i)erson  has  a  hours  at  his  disposal ;  how  far  may  he 
ride  in  a  coach  which  travels  b  miles  ])er  hour,  and  yet  have 
time  to  return  on  foot  walking  c  miles  per  hour  ?  See  Ex. 
48.  Art.  224. 


252  TEXT-BOOK    OF   ALGEBRA. 

9.  A  can  do  a  piece  of  work  in  j)  days,  B  in  q  days,  and 
C  in  r  days.  In  how  many  days  will  they  finish  it  all 
working  together  ?     Given  p  =  10,  2'  =  15,  r  =  30. 

10.  The  sum  of  two  numbers  is  equal  to  a  and  their  sum 
is  to  their  difference  as  m  is  to  n.  Required  the  numbers. 
Given  a  =  17,  m  =  6,  ?z  =  5. 

11.  Divide  the  number  n  into  two  such  parts  that  the 
quotient  of  the  greater  divided  by  the  less  shall  be  q  with  a 
remainder  r.     Put  n  =  485,  f^-  =  79,  r  =  5. 

12.  Divide  the  number  d  into  three  such  parts,  that  the 
second  shall  exceed  the  first  by  h,  and  the  third  exceed  the 
second  by  c.     Put  d  =  588,  b  =  64,  c  =  91. 

13.  A  and  B  together  can  perform  a  piece  of  w^ork  in  d 
days,  A  and  C  together  in  e  days,  and  B  and  C  together  in 
/days.  In  what  time  can  each  person  alone  perform  the 
work  ? 

14.  A  merchant  bought  p  gallons  of  two  kinds  of  oil  giv- 
ing for  one  m  cents  a  gallon  and  for  the  other  7i  cents  a 
gallon.  By  mixing  them  and  selling  the  mixture  at  r  cents 
a  gallon  he  gained  $a.  How  many  gallons  of  each  did  he 
buy? 

15.  In  an  old  Cliinese  arithmetic  called  Kiu  Tschang, 
which  was  comjileted  about  2(300  b.c.  and  elucidated  and 
enlarged  about  1250  a.d.  by  Tsin  Kiu  Tshaou,  there  is  found 
the  following  problem:  In  the  middle  i)oint  of  a  square 
l)ond  m  feet  each  way,  there  grows  a  reed  which  rises  p  feet 
above  the  water.  Now  if  the  reed  be  pulled  to  the  middle 
])()int  of  one  of  the  edges  of  the  pond  it  just  reaches  to  the 
to])  of  the  water.  What  is  the  depth  of  the  water  ?  See 
Ex.  88,  Art.  224. 

1.)    Vutm  =  10,^  =  1. 

2.)    One  side  of  a  right  angled  triangle  is  25  inches,  and 
the  other  increased  by  12  inches  equals  the  hypothenuse. 


SIMI'LK    Kl^LATlnNS.  253 

What  is  the  hypothenuse  equal  to  ?     Solve  by  substitution. 
What  does  m  =  ,  what  j)  ? 

3.)  One  side  of  a  rectaus^nilai  tract  of  hind  is  one  niik*, 
and  the  other  side  increiised  by  .414  mile  is  ec^ual  to  the 
diagonal.  What  is  the  length  of  the  diagonal,  and  what  is 
the  length  of  the  other  side  ?     What  does  p  equal  here  ? 

16.  If  the  selling  price  of  a  certain  product  is  p  a  mer- 
chant gains  71%.  How  many  per  cent  is  gained  or  lost  if 
the  selling  price  is  />'  ? 

17.  Two  bodies  move  towards  each  other  from  points  I 
meters  apart.  The  one  p  meters  a  minute,  the  other  q 
meters.  In  how  many  minutes  will  they  be  r  meters  from 
each  other.     I*ut  /  =  500,  j)  =  30,  fj  =  2r>,  ;•  =  00. 

18.  The  fore  wheel  of  a  wagon  is  a  feet  and  the  hind  wheel 
b  feet  in  circumference.  Through  what  distance  must  the 
wagon  pass  in  order  that  the  fore  wheel  shall  have  made  n 
more  revolutions  than  the  hind  wheel  ? 

19.  A  person  distributed  a  cents  among  n  beggars,  giving 
b  cents  to  some  and  c  cents  to  the  others.  How  many  were 
there  of  each  ? 

1.)  A  father  divides  $8500  among  7  chihlren,  giving  to 
each  son  $1750,  and  to  each  daughter  $500.  How  many  of 
his  children  were  sons  and  how  many  daughters? 

20.  A,  B,  and  C  hold  a  ])asture  in  common  for  which  they 
l)ay  q  dollars  a  year.  A  puts  in  a  cows  for  m  months,  B,  b 
cows  for  n  months,  C,  c  cows  for  p  months.  Required  each 
one's  share  of  the  rent. 

1.)  q  =  $181.20,  a  =  6,  ^  =  5,  c  =  8,  m  =  30,  7j  =  40, 
JO  =  28. 

20'.  A  boy  who  had  three  studies,  Latin,  Greek,  and 
mathematics,  received  at  the  end  ot  a  certain  term  95  in 


254  TEXT-BOOK    OF    ALGEBRA. 

mathematics,  90  in  Latin,  and  85  in  Greek.  The  mathe- 
matics recited  5  times  a  week,  the  Latin  4  times,  and  the 
Greek  3  times.  What  was  his  average  for  the  term  in  all 
his  studies  ? 

To  make  a  convenient  formula  for  calculating  an  average 
grade  a*,  let  a  be  an  approximate  value  of  the  average  grade 
(as  90,  or  80,  or  60),  and  let  Z,  w,  n  be  the  differences  (taken 
with  proper  signs)  between  the  given  grades  and  the 
assumed  average  grade.  Also  let  ^,  c,  and  d  be  the  number 
of  hours  per  week  the  respective  studies  recite.     Then 

X  =  (^-^  +  0  ^  +  ((f'  +  fii^)  e  -{-  (a-\-  n)  d  __  ^^   .    lb  -^mc  -\-  nd 
h  -^c-{-  d  '^      h  j^c^  d 

which  expression  is  more  convenient  for  calculating  the 
value  of  X. 

21.    To  derive  the  rule  for  multi[)lication  of  fractions. 

a         c 
Let  y  and  -.  be  the  factors  and  x  their  product. 


Then  x  =  ~fX-j 
h      d 

a            e 
hdx  =  j^-bX~^-d 

(Ax.  .3  and  38,  2) 

=  a  X  c 

(by  definition  of  division,  43) 

(Ax.  4) 

a      c       ac 

^^^'    h^d  =  M 

Hence,  to  multiply  fractions  multiply  their  numerators 
for  a  new  numerator,  and  their  denominators  for  a  new  de- 
nominator. The  result  is  then  to  be  reduced  to  its  lowest 
terms,  by  striking  out  equal  factors  in  the  numerator  and 
denominator.  To  save  unneQessary  writing  this  cancellation 
is  done  before  and  not  after  the  terms  are  multiplied. 


S!.MIMJ<:    KQUATIONS.  255 

22.    'l"u  UfiiM'    ihf    nilt^    i'or  divisiuu   oi    iiiuLion>.      [a-I 

__  ami  —  be  the  tractions  and  x  their  quotient.     Then 
b  (I 

(I         c 
c  ^  d         f a       c\      e       d 

But  l)v  tlic  (l(»tiuiti()n  of  division  (  -  -^  -]X   ,  =  ,  ,  and 

\0       a  J      d      0 

c       d 
by   the  preceding  ^  X  7  =  1 .     Then 

a      d 

, V       (t        r  "  ^d  ad 

Hence,  to  divide  one  fraction  by  another,  iiivtTl  tlui 
divisor  and  multiply. 

Kkm AKK.  —  It  was  thought  that  the  student  would  be  able  to 
understand  these  proofs  better  after  studying  equations.  They 
might  have  been  given  just  as  they  stand  in  articles  178  and  180. 
They  include  as  particular  cases  the  principles  of  division  often 
given : 


,,....        ,  S  dividend  ,.  . ,.         ,  S  divisor 

Multiplying  the  J  ^^^^^^ator  «^  ^^'^'^'^^  the  ^  ,,,.„,,,„i„ 

....       ,  S  quotient 

multiplies  the  J  ^,^,^^^  ^^  ^,^^  ^^^^.^^^  . 

, S  dividend  ,  .   ,  .        ,  S  <^i 

Dividing  the  ^  n„,„erator  ^'  "»"ItM>ly.ni:  tlu-  J  ^,^, 


itor 


divisor 
'nominator 
divides  the  fraction. 


THIRD  GENERAL  SUBJECT. 

(THE  NOTATION   CONCLUDED.) 

POWERS,  ROOTS,  AND  RADICALS. 


CHAPTER   XVII. 

OF    POWEU8. 

256.  Involution  as  a  term  in  algebra  signifies  raising 
qnantities  to  powers. 

a.  Involution  is  merely  one  case  in  multiplication  (56,  67)  where 
the  numbers  multiplied  are  equal.  And  so  some  exercises  in  involu- 
tion have  been  given  under  multiplication  (117).  For  present  pur- 
poses it  will  be  convenient  to  treat  this  subject  under  three  heads  : 
monomial  powers,  binomial  powers,  and  polynomial  powers  of  three 
or  more  terms. 

SECTION  I. —Monomial  Powers. 

257.  Exercise  in  Involving  Monomials  to  Powers.  —  The 
student  should  familiarize  himself  anew  with  117  and  128. 

a.  To  develop  a  fraction  both  terms  nuist  be  raised  to  the  proper 
power.     (178.) 

4        7  a"* 
3^V'      2~J 


1.    Square  «'V,  11  Ire',  —  4  a%^x',  —  |  trx'^,  —  o— g- 


""'h'%  (^l~~j  ,  on-\  2-^  •  2-^  (3|)-^  (-  a'xy,  (-  af  (-  by 

93      9 

(-  C)\  ^5,  %  a«j"«+-;  yP-\  yU-X  ym^l 

256 


POWEKS,    HOOTS.    AND    RADICALS.  257 


2.    CiiIk',   —  0  «•,   a'\   h^if,    n^^  X  a- 


--  e;:j 


a) 


NoTK.  —  111  tlio  last  (!Xi>ressiou  it  is  not  known  whether  lu  is  oven 
or  odd,  and,  consequently,  wliether  the  result  is  +  or  — .  In  such 
cases  (—  1)  "•  may  be  written  as  the  si(jn  coefficient. 

4.  Kaise  —  ((ibr)'^  to  the  Hftli  i)o\\t'r:  niise  —  crb^c^  to  the 
With  power,  when  m  is  even. 

5.  Find  the  cube  of  (—  >i/*\-'-}  (—  ^/•^^V") ;  the  square  of 
4a*b*c  (—  xy.-v*) ;    the  fifth  \n)\\(tv  of  —  "^  x" ,  the  cube  of 

—  -! ;  the  seventli  power  of  abc^ ;  the  sixth  power  of 

i>    b~^ 


(?)'{f)" 


6.    Simplify  2  a  (—3  ft2,/iwy;   (4a^")';  o  (W)"  ;  a{5nf\ 

7  (7  pqh-y ;   3  ( -  2  <//;)-- :    ( _  4)^  X  6  (^0' ;   ^'  ("'"^'"? ; 

(ab-^c--)-^  ;   (x'y-')-  ;   (^)^}^ '    ("  *)""^  ><  ("  1)""  ' 


"8   •"  '^    "^ 


2rV/Y 
.•fa/icj 


SECTION  II.  —  Binomial  Powers. 

258.  Investigation  of  Binomial  Powers.  —  xV  study  of  the 
development  of  binomials  leads  to  Xewton's  theorem. 

The  development  of  the  square  of  a  binomial  was  studied 
in   Thr'nnMns   [.  and    IT.  in    nmlti]>lifation  ;   then  the  other 


258  TEXT-BOOK   OF    ALGEBRA. 

simple  cases  of  the  cube  and  the  fourth  power  of  a  binomial 
in  116,  4  and  6.  But  each  of  these  led  to  a  special  theorem. 
We  are  now  to  seek  for  a  general  law  applicable  alike  to  all 
powers. 

Let  A  and  B  represent  any  two  quantities.  Then  A  -|-  B 
or  A  —  B  is  a  binomial.  We  proceed  to  form  the  powers 
by  multiplication. 

A  +B 
A  +B 
A^  +  AB 

+  AB  +  B^ 

A2  +  2  AB  +  B2 
A  +     B 


A^  +  2  A^B  +  AB-^ 

+  A^B  +  2  AB^^  +  B^ 
A«  -h  3  A43T3  AB^  +  B8 
A  +     B 


A^  +  3  A^B  +  3  A^B^  +  AW 

+     A^B  +  3  A^B^  +  3  AB«  -f  B^ 
A*  +  4  A^B  +  6  A^B^  +  4  AB^  +  B* 

A  +     B 

A«  +  4  A*B  +  6  A«B2  +  4  A^B«  +  AB^ 

+     A^B  +  4  A^B'^  +  6  A^B^  +  4  Al^^  +  B^ 


A*5  +  5A^B  +  10A3B2  +  10A=^B«  +  5AB*  +  B5  =  (A  +  B)s 


A  -B 
A  -B 
A^^^^TB 

-  AB  +  B^ 
A^  —  2  AB  4-  B2 

A  —     B 

A3  —  2  A'^B  +  AB^ 

-  A^B  +  2  AB^  -  B« 

A3  —  3  A^B  +  3  AB^  —  B^  =  (A  —  B^^ 

A  -     B         

A*  —  3  A«B  +  3  A'^B'  —  AB^ 

-  A^B  +  3  A^B^  -  3  AB«  +  B^ 
A^  —  4  A^BTTA^B^  —  4  AB^  +  B^ 


POWERS,    ROOTS.    AM)    KADK    \LS.  259 

A  careful  sciutiny  of  tliese  iiiiiltiplicatioiis  leads  us  to 
the  following  eonclusions.  whicli  if  true  in  all  the  above 
cases  we  may  guess  to  Ix-  tun  d  all  powers.  The  (h'nn^n- 
sf ration  of  Newton's  theorem  which  is  rather  too  difficult 
to  be  inserted  here  does  justify  the  conclusions  about  to  be 
given. 

In  tlie  development  of  a  binomial,  as  (a-\-b)"y  we  learn 

1.  The  number  of  terms  is  always  one  greater  than  the 
exponent  of  the  power  to  which  the  binomial  is  raised. 

Thus  there  are  five  terms  in  the  fourth  jwwer. 

2.  The  exj)onent  of  the  leading  letter  in  the  first  term  of 
any  jK)wer  is  the  same  as  the  exponent  of  the  })ower  of  the 
binomial.  In  the  second  term  it  is  one  less,  in  the  third 
term  one  less  than  in  the  second  term,  and  so  on,  the  lead- 
ing letter  not  appearing  in  the  last  term.  On  the  other 
hand  the  other  letter  begins  in  the  second  term  with  the 
exi)onent  1  which  increases  by  unity  each  time,  until  in 
the  last  term  it  is  the  same  as  that  of  the  power  of  the 
binomial.  Thus  the  product  is  symmetrical  with  respect 
to  the  letters. 

3.  When  the  binomial  is  a  sum  the  signs  of  all  the  terms 
are  ]X)sitive.  "But  when  the  binomial  is  a  residual,  every 
term  which  contains  an  odd  power  of  the  second  letter  is 
minus. 

4.  llu'  coefficient  of  the  iirst  t* mi  is  1  (understood),  as 
also  that  of  the  last  term.  The  coefficient  of  the  second 
term  is  the  same  as  the  exi)onent  of  the  power  of  the  bi- 
nomial, as  also  that  of  the  next  to  the  last  term.  Further- 
more, the  coefficients  increase  to  the  middle  term,  or  terms, 
and  then  diminish  to  the  last,  those  equally  distant  from 
the  extreme  terms  l>eing  ec^ual  to  each  other. 

5.  The  third  Coefficient  can  always  be  derived  from  the 
second  by  multiplying  the  second  by  1  less  than  itself  and 


260  TEXT-BOOK    OF   ALGEBRA. 

dividing  the  product  by  2.  The  fourth  coefficient  may  be 
derived  from  the  third  by  multiplying  it  by  1  less  than  the 
previous  multiplier,  and  dividing  by  1  greater  than  the 
previous  divisor  (i.e.  by  3),  and  so  on,  the  multiplier  be- 
coming 1  less  and  the  divisor  1  greater  than  the  last  for 
every  new  term.  Thus  the  coefficient  6  in  the  third  term 
of  (A  -f-  ^y  is  derived  from  the  4  by  multiplying  4  by  4  — 
1  =  3j  and  dividing  by  2  ;  the  coefficient  4  following  the  6, 
by  multiplying  6  by  2  (1  less  than  3,  the  previous  multiplier) 
and  dividing  by  3  (1  greater  than  2,  the  previous  divisor). 
The  last  coefficient,  1,  from  the  preceding  coefficient,  4,  by 
multiplying  it  by  1,  and  dividing  the  product  by  4. 

Unlike  the  preceding  laws,  the  fifth  is  probably  too  com- 
plex for  the  student  to  have  perceived  it  unaided. 

a.  There  is  a  memoriter  rule  which  can  be  more  easily  followed 
when  A  and  B  stand  for  two  first  powers :  —  Multiply  each  coeffi- 
cient by  the  exponent  of  the  leading  letter  in  that  term  and  divide 
the  product  by  the  number  of  that  term  from  the  left.  The  quo- 
tient will  be  the  coefficient  of  the  next  term. 

Thus  in  the  fifth  term  of  (A  +  B)5,  5  AB*,  the  5  is  found  from 
10  A^B^  by  multiplying  10  by  2,  the  exponent  of  A,  and  dividing  the 
product  by  4,  the  number  of  the  term  10  A-^B=^  from  the  beginning. 

It  may  be  observed  that  the  divisor  is  always  1  greater  than  the 
exponent  of  the  second  letter. 

b.  The  student  will  find  it  advantageous  to  remember  the  bino- 
mial coefficients  up  to  the  fifth  or  sixth  powers. 


259.    Examples  of  the  Use  of  the  Binomial  Theorem. 

1.    Develop  (A  -|-  B)^  by  the  theorem. 

In  this  expansion  there  will  be  6  terms,  and  hence  when 
three  coefficients  are  found  the  others  can  be  written  down 
directly,  being  those  already  obtained  in  reverse  order.  The 
reasons  in  parentheses  refer  to  sections  in  the  previous 
article. 


POWKHS.    KOOTS,    AM)    i;\l>l("AI>S.  261 

(4,-.  (4,2)  ^     ,...  ura,  .;  (4,2) 

(A  -h  ])f  =  \'  -h  r>  An5  -f  '^"-^  A«B'^  +  10  A'^B» 

(4, -'J         l4.  I.') 

-f  5  AB^+  li^ 

2.  Develop  (x  —  y)®. 

Here  there  will  be  9  tei-ms  in  all,  and  it  will  be  necessary 
to  determine  5  coefficients. 

(.1,4)  (5,  or  a)  (3,4) 

(^  _  ,/).  =  ..*  _  8  ^'y  +  ^  (=  28)  ^y  -  ?5^ 

(3.j)ro)  (4) 

(=  56)  xY  +  '^^^  (=  70)  j'Y  -  5(>  :ry  +  28  xY  -         ' 
8a-y  +  / 

3.  Develop  (2  x'^  —  3  (///)*. 

Here  A  =  2a;^,  and  !>  .'I'/y.  Fo  preserve  their  iden- 
tity they  are  written  in  parentheses,  treating  (2  x^)  as  A, 
and  (.3  «y)  as  H. 

4X3 

(2  «*  -  3  at/y  =  (2  x'Y  -  4  (2  j--^)«  (3  «y)  +  -^—  (2  x^y 

(3  ay)^  -  ^'  3-    (2  o-'^  (3  «y)«  +  GJ  ^^V)* 

The  single  terms  must  now  be  simplified. 

(2  xY  =  10  .rs .  4  (2  X'') »  (3  .^y)  =  1)0  «.rV ;    -^  (2  a;^« 
(3ay)''^=216«*V/ 
5A?  (2  ««)  (3  fly)"  =  21  (;  ^/ 1/  -//' :    ( .;  ^/y)*  =  81  ay.     Hence 

(2  J--  -  3  «y)*~16  x«  -  m  ux''i/  +  216  a^y  -  216  a»«y 
-f  81  aY- 

Note.  —  The  student  must  remember  to  keep  distinct  the  bino-  * 
mini  coefficients,  and  any  numbers  that  may  come  into  the  result 
from  numeriral  coefficients  in  tlie  quantities  developetl.  We  could 
never,  for  instance,  obtain  the  third  coefficient  210  in  the  expansion 
just  given  by  multiplying  90  (!)  by  «.')  and  dividing  by  2.  For  the 
binomial  coefficient  of  that  term  is  not  00  but  4. 


262  TEXT-BOOK   OF   ALGEBRA. 

260.    Exercise  in  Raising  Binomials  to  Powers. 


1. 

(J99.  -  ^'Y- 

10. 

(2  r  —  6  my. 

2. 

{2  m— pf. 

11. 

(la-?,  by. 

3. 

(x'-^4.,fy- 

12. 

(5  ic5  _  4  ^4^); 

4. 

{5  a  —  bey. 

13. 

(a  -  ly.  ■ 

5. 

(2  ^2  +  axy. 

14. 

(§  ^  -  fl  yy- 

6. 

(1-2  by. 

15. 

ix'-  -  2  xi/y. 

7. 

(a  +  iy. 

16. 

8. 

(         2^V 

17. 

{x^  -  2y. 

9. 

io-^xy. 

18. 

{x-^  -  Iff. 

19.  Find  the  first  four  terms  of  (2  am  -\-  ly. 

20.  Verify  99^  =  (100  —  1)^ 

21.  Cube  by  the  theorem  999  =  (1000  —  1). 

22.  Raise  9999  to  the  fourth  power. 

23.  Raise  12^  =  12  +  i  to  the  fourth  power. 

24.  Raise  1892  =  (1900  —  8)  to  the  third  jxnver. 

SECTION  III.  —  Polynomial  Powers. 

261.  Development  of  Polynomial  Powers. — rolynomials  are 
most  readily  developed  by  regarding  them  as  Binomials, 
and  using  Newton's  theorem. 

a.  Polynomial  squares  are  easily  written  out  by  the  rule  given  in 
116,  3. 

1.    Required  to  cube  a-\-b-\-c. 

To  make  a-j-b  -\-  c  a  binomial  it  is  written  {a-{-b)  -]-  c. 
Next  the  binomial  thus  formed  is  developed,  and  afterwards 
the  different  powers  oi  a-\-b  are  expanded.  Last  of  all  the 
resulting  indicated  opei-ations  are  performed. 


POWEHS,    tcfM.i>.    A  Mi    KADICAL8.  263 

By  section  II.  we  have 

[(«  +  b)  +  cf  =  (a  +  by  +  :;  i  '/  +  h)-r  +  o  yn  +  h)c-^  -f  c* 

=  a"*  +  3  aV,  +  3  ab'^  -{■  h^  +  :'»    -.' '  +  i'  ab  +  b'^)r.  +  3  (a  +  6)c2  +  c» 

=  ^3  +  3  (I'b  +  3  «6-  +  /;•»  +  ;i  r/-r  +  r.  f/^f  +  3  b-^r  +  3  ar-2  +  3  bc^  +  c« 

^1  tlH. 

=  a-^  +  /*■»  +  6-84-  3  (a^/)  4-  a-r  +  ab'^  +  ac-  +  b-r  +  ^f-^)  +  6  rr^r. 
State  this  result  in  the  form  of  a  theorem. 

2.    Required  the  fourth   power  of  'An  -|-  2  A  —  r -f       . 

[(3a  +  '2b)  -  /<•  -  ^\  J*  ^  (3a  +  2 />)*  -  4  (3  a  +  'J  b)^   U-  |\ 

+  (J  (3  a  +  2  />)-•  ^-  -  '-'"j  '-  4  (3  a  -\-2b)    (''-  tY 

=  (3  a)*  +  4  (3  a)3  (2  6)  +  6  (3  a )2  (2  by  +  4  (3  a)  (2  by  4-  (2  b)* 
-  4  [(3  ay  +  3  (3  ay  (2  6)  +  :;  (:5  (0  (-^  ^)*-^  +  (-'  '^)'  ]    [^  "  ^  ] 
+  C.  fna-i  +  12  rt/>  +  4  />-'"]    fc-^  -  C(I  4-  ^1 
-4[3a  +  2/>]   [c^-3r:^  2  + -^  ^"  T  "  S  J 

rpoii  developing  tli«'  iii<»n<>iiii;il  txpi.  -^i(.ll■^  and  ix'rforniinii  the 
imdtiplications. 

[:;  a  f  2  /»  -  /•  -f  I)    -  81  a^  +  2l»;  a'/,  4-  2ir,  a-7/-!  4-  •.»«;  afri  +  ir.  M 

-  108  a-^r  -  2 Hi  a-^r 
-  144  at^r  -  32  />^r  +  54  a^d  +  108  a-bd  +  72  a^tZ 
4-  K^  Ifld  +  54  rt-^f'-J  4-  72  abr^  +  24  fi^r^  -  54  a^rvi 

_  -JO  ,,}„',!  -  24  //-i^-f?  4-  —  «2'''-  +  1><  "/"'-'  +  <•'  ''■-'?■-  -  12  ac» 
4-  ]>«>  (/'-■'/  —  1*  ard^  +  -^  ar/i  —  .s  /k--  -t-  i:.'  hr-d  —  tj  ftccZ^ 
4-  biV'  4-  r*  -  2r«/Z  4-  :;  ^-'?-  -  :;  rfZ»  4-  -^  ■ 


264  TEXT-BOOK   OF   ALGEBRA. 

3.  Square  xy  +  yz  -\-  zx. 

4.  Develop  (1  +  x  +  x'  +  x^. 

5.  Cube  a  -\-  b  —  r. 

6.  Develop  (1  —  a  —  a^)^ 

7.  Develop  (1  —  2  x  -\-  x^. 

8.  (2  rt»*  -  b''  -h  cr^y  =  ? 

9.  Cube  ax  +  />//  +  1. 

10.  Develop  (a  -^  2  h  —  c)*. 

11.  Develop  (1 -{- x  +  x^. 

12.  Develop  (ax  +  /;//  +  1)^ 

13.  (x-{-y-{-iy  =  ? 

14.  Cube  1799  =  1000  -f-  800  —  1„  and  verify  the  answei 


I'oWl.KS.     iMtoTS,    AN1>     1!  A  1  >I<  "A  I.S.  265 


CHAPTER   XVIII. 

OF  EXACT   ROOTS. 

262.  Evolution  iis  ;i  tt'ini  in  algebra  signifies  extracting 
the  roots  of  quantities. 

<t.  This  chapter  will  be  separated  into  the  following;  sections: 
roots  of  monomials;  the  square  root  of  polynomials;  the  ciil>e  root 
of  polynomials;  other  roots  of  polyiiomials. 

SECTION  I.  — Monomial  Roots. 

263.  Roots  of  Monomials.  —  As  in  simple  multiplication 
and  division,  there  are  three  thinj^s  to  be  considered  :  the 
coefticient.  the  sign,  and  the  literal  ])art.  Consult  articles 
45.  46.  and  129.     See  also  58,  59,  60. 

264.  Signs  of  Roots.  (45.)  The  reason  for  the  existence 
of  two  scpiare  roots  so  different  in  character  may  at  first 
puzzle  the  beginner.  It  is  a  consequence,  however,  of  the 
rule  of  multiplication  that  minus  by  minus  as  well  as  plus 
by  plus  gives  ])lus.  I*n)p('r1y  s])eaking,  the  sign  -j-  slionld 
l)e  read  "  jdus  ami  minus."  inst-ad  of  "  plus  or  minus.'"  1 1, 
now,  it  is  known  that  a  power  has  been  produced  by  multi- 
plying two  positive  numbers,  then  its  root  is  positive;  or, 
if  by  two  negative  factors,  it  is  negative.  In  tlie  solution 
of  arithmetical  iiroblems.  the  nature  of  the  i)roblem  may 
admit  of  only  positive  aiisuiMs:  tln'ii  only  the  positive  root 
is  taken  for  the  answer. 

265.  Imaginary  Quantities.  —  It  can  be  shown  that  a  real 

meaning  may  be  assigned  to  what  have  been  called  '' imagi- 


266  TEXT-BOOK    OF   AUilCBllA. 

nary,"  or  "impossi})le  "  quantities  (45).  But  it  is  only  by 
going  out  of  the  realm  of  algebra  proper  that  it  can  be 
done.  Whenever  an  imaginary  value  is  found  for  the  un- 
known in  a  problem,  it  shows  that  the  problem  is  algebra- 
ically impossible,  just  as  a  negative  result  shows  its  problem 
to  be  arithmetically  impossible. 

266.    Exercise  in  Extracting  the  Real  Roots  of  Monomials. 


1-    ^a%\  -<Jx%  ^Jc"^,   V— 4««/^--^,   V— 64,   VSl^^'^ 
V27;  V.?%  ¥25^   ^         'V         121,   V^ 


81 


36  h'^ 


3.  Find  the  cube  root  of  ^"'^  ^  ^  ;  of  "^  ^^^. 

216  ;ry'  8 

4.  *  /  _  343  ^12 ^9^  . /_  -^^-      V-  a'lf'c\  v/-  16a«a'i-^. 
V  V        64?/^^,  V 


c^d^'x^ 


8.  Express  the  n*''  root  of  x,  of  7. 

9.  Extract  the  roots  and  simi)lify 

10.  ^^_  21  Wf  X  V243^^^  X  VUhr^  ==  '' 

SECTION  ir.  —Square  Root  of  Polynomials. 

267.    Extraction  of  the  Square  Root  of  Polynomials.  —  How 

effected. 

a.    If  the  square  root  of  a  polynomial  is  not  a  binomial  it  may 
readily  be  put  in  the  form  of  one  (as  in  261).     Now,  just  as  the  bino- 


POWERS,    ltOOT!=>,    AND    KADK  ALS.  207 

mial  theorem  would  serve  to  develop  the  powers  of  any  polynomial, 
so  here  the  investigation  of  hinomial  roots  will  include  the  general 
case. 

1.  Required  to  extract  iIk  Sipiare  root  of  A^  J:;  2  AB  +  B'^. 
Referring  to  258  we  see  that  when  a  power  Is  arranged  the  root  of 

its  first  term  is  the  tirst  term  in  the  root.  Hence,  in  extracting  roots 
in  order  to  make  a  heginning,  the  quantity  is  arranged  with  refer- 
ence to  the  exponents  of  its  leading  letter,  and  the  root  of  its  first 
term  is  then  extracted. 

Explanation.  —  The  first  term  of 
A2  +  2  AB  +  IP     (A  +  B     ^j^g  ^^^^  ^  having  been  found,  as  just 

„ 1 --  explained,  it  remains  to  find  the  otlier 

2  A  +  B I  2  AB  +  B-^  ^^^^^^  ^^  ^^^^  ^.^^j^.  ^^^^^j  ^j^^^^  j^.,  j^  ^j^^ 

'  square  of  A  alone,  A  is  squared  and 

subtracted  from  the  polynomial,  leaving  2  AB  +  B^  =  (2  A  +  B)B. 

Of  course,  in  this  example,  w  know  perfectly  well  that  the 
square  root  of  vV^  ±  2  AB  +  B-^  is  A  ±  li.  (See  the  tlieorems  in  mul- 
tiplication.) Hence  B  is  the  second  term  of  the  root  sought.  Man- 
ifestly, now,  if  2  AB  is  divided  by  2  A  (double  the  root  term  alrea«ly 
found),  B  is  the  quotient.  Similarly  —  2  AB  ^  2  A  =  —  B,  which  is 
the  second  term  of  the  root  in  this  case. 

Lastly  we  observe  that  if  B  be  now  annexed  to  2  A  and  the  sum 
multiplied  by  B,  the  product,  2  AB  +  B^^^  subtracted  from  the  previ- 
ous remainder  leaves  zero,  and  the  process  is  complete. 

To  show  how  this  method  applies  wlien  there  are  more  than  two 
terms  in  the  root,  let  us  take  the  s(|uart'  of  a  +  h  +  c  =  a^  +  2ab  + 
//-  +  2  ac  +2  he  +  cr  which  can  \n-  w  i  itten  (a  +  6)2  +  2  («  +  6)  r 
+  '-.  Thus,  when  the  first  two  t.  iiii>.  d  +  6,  are  given,  the  third 
can  1m'  found  l)y  dividing  tin-  lir>t  tnni  of  the  remainder,  2  (r/  +  />)  r, 
by  twice  the  root  already  found.  Now  the  root  terms,  a  +  h,  must 
be  found  from  the  first  terms  of  the  polynomial  in  a  preliminary 
ojieration  similar  to  that  given  above  for  A  +  B,  and  its  S(iuare  sub- 
tracted leaves  the  remainder  2  («  +  6)  r  +  c'-.  If  there  are  still  other 
tenns  in  the  root,  t6  find  each  succeeding  term  all  of  those  already 
found  are  considered  as  one  quantity.  To  better  illustrate  this,  we 
give  another  example: 

2.  Required  to  extract  the  square  root  of  Ax"^  +  n'^if 
-*-  9  .t;*  —  4  axil  +  1 2  -rz^  —  i\  oi/x^. 


268  TEXT-BOOK  OF  ALGEBRA. 

Arranging  this  with  reference  to  the  powers  of  x  (and  y  and  2), 
we  have 

4:  x^  —  4c  axy  +  12  xz'^  +  a^y-  —  0  ayz-  +  9  z^  ((2  x  —  ay)  +  3  z'^ 
4x^ 


4x  —  ay 


—  4  axy  +  12  xz^  +  ahj^ 

—  4  axy +  ary'^ 


4x  —  2ay  +  Sz^    12  xz^  —  6  ayz'^  +  92* 

12  xz^ -  6  ayz^  +  9  2* 

In  this  problem  after  two  terms  of  the  root  are  secured,  they  are 
considered  as  one  quantity  and  doubled  for  tlie  trial  divisor.  Divid- 
ing we  get  the  third  term  of  the  root,  which,  when  found,  is  an- 
nexed to  the  others  with  its  proper  sign.  And  so  in  all  cases,  the 
root  already  found  is  treated  as  one  quantity. 

268.  Rule  for  Extracting  the  Square  Root  of  Polynomials. 

1.  Arrange  the  polynomial  with  reference  to  its  leading 
letter,  extract  the  square  root  of  the  first  term,  and  subtract 
its  square  from  the  polynomial. 

2.  Divide  the  first  term  of  the  remainder  by  double  the 
root  quantity,  and  the  quotient  will  be  the  second  term  of 
the  root,  which  is  written  at  the  right  of  the  previous  or 
trial  divisor,  as  well  as  in  the  root.  This  complete  divisor 
is  then  multiplied  by  the  root  term  just  found  and  the  pro- 
duct taken  from  the  remainder. 

3.  Double  the  two  terms  of  the  root  for  the  next  trial 
divisor  and  annex  to  it  the  new  root  term,  when  found,  be- 
fore multiplying.  Continue  the  operation  until  there  is  no 
remainder,  or  as  far  as  desired. 

269.  Exercise  in  Extracting  the  Square  Root  of  Polynomials. 

1.    W  -  2  AB  +  A2  (  =  A^  -  2  AB  +  B"^). 

The  root  is  B  —  A,  which  differs  in  sign  only  from  A  —  B, 
the  root  previously  obtained.  Thus,  B  — A  =  —  (A  —  B). 
(See  45,  4,  (2.)  Moreover,  this  is  true  of  all  polynomial 
square  roots. 

If  a  polynomial  is  a  square  root  of  a  quantity,  the  same 
polynomial  with  all  of  its  signs  changed  is  also  a  root. 


POWKItS,    i:()(/rs,    AND   RADICALS.  JOU 

2.   /*  +  r)/V  4-  \)j\ 

4  ^9 

4.    fi-^  -  2  +  </. -2. 

Kkmauk.  —  Trinomial  squares  can  usually  hv  n'solvod  mentally, 
a-'"  -I-  2  a'"ic"  +  07"*"  ' 

8.    ^/a*  -I-  8  a«i  +  24a'^^»2  _^  39  ab^  +  16  //. 
Suggestion.  —  Extract  the  square  root  twin'.     (^Se»'  46,  :>.) 

'■('+:)'-('+'-' 

10.  1  -2,v  +  -^^-^«-|-^'. 

11.  ./•-  -  2  ./•  4-  1  +  2  .,•//  -  2  //  -f  //■-. 

12.  .r(.r-f-l)(.r-h  2 )(./•  +  ;;)  +  1. 

13.  (13  r-^)-  -f-  (4  .r')-  +  (7  .!•)-  +  2H>  .r'  -  12(Lr^ 

14.  Obtain  four  terms  of  ^/„i  _j_  y\ 

n^^'       bn^       b'-n'^~''       n"^'-       hit''       tr 
'  \?~  "•"  G^  +  ~T44'  +    ./     +  12  +  4' 

270.  Extraction  of  the  Square  Root  of  Arithmetical  Numbers 
as  Polynomials.  —  Tlic  lulc  for  tlu*  extraction  of  iiuiiil)'is 
('written  in  the  Arabic  notation)  is  a  sjiecialized  form  of  tlmt 
for  i)olynomial.s.     We  jiroceed  to  illustrate  this  by  e.xamples. 

1.    Required  to  extriu-t  the  square  loot  of  :{44.")(;<.). 
We  write  the  number  as  a  ]>olynomial  whose  terms  li;i\  <• 
resj)ectivrly  an  even  number  of  ciphers. 


270  TEXT-BOOK   OF   ALGEBRA, 


^40000  +  4500  +  69  |500  +  80  +  7 
250000 

1000  +  80 
(1080) 

90000  +  4500  =-  94500 
8()400 

1160  +  7 

(1167) 

8100  +  69  =  8169 
8169 

Explanation.  —  Starting  at  the  decimal  point  the  numher  is 
separated  into  terms  of  two  tigures  each,  annexing  an  even  number 
of  ciphers  to  fill  out  the  vacant  orders.  Evidently  two  ciphers  in 
the  power  will  give  one  in  the  root,  four  in  the  power  will  give  tw^o 
in  the  root,  and  so  on.  This  enables  us  to  get  the  first  figure  in  the 
root.  For  the  first  term  of  this  number  polynomial  Mill  give  the 
first  term  in  the  root.  In  the  present  example,  the  first  figure  of 
the  root  cannot  be  (>  for  600  squared  is  360,000,  which  is  greater 
than  the  given  number  344569.  Taking  500  as  the  greatest  number 
of  hundreds  and  subtracting  its  square  from  the  first  term  gives 
9000,  which  added  to  the  next  term  makes  94500. 

Doubling  the  root  found,  500,  the  trial  divisor  is  found  to  be 
1000.  This  is  contained  in  the  remainder  90  times;  but  upon  add- 
ing 90  to  the  trial  divisor  and  multiplying  by  90  the  product  is 
98100,  which  is  too  great.  Hence  eighty  is  the  second  term  of  the 
root  instead  of  90.  Writing  80  in  the  root  and  adding  it  to  the  trial 
divisor,  we  have  1080  for  the  complete  divisor.  Multiplying  the 
complete  divisor  by  the  second  term  of  the  root  and  subtracting  the 
product  from  94500  the  'remainder  is  8100,  to  which  the  last  term  of 
the  polynomial,  69,  is  added. 

Doubling  the  root  already  found  we  have  for  a  new  trial  divisor 
1160,  from  which  7  is  found,  and  the  process  is  completed  by  adding 
7  to  the  trial  divisor  and  multiplying.  The  500,  of  course,  corre- 
sponds to  A,  and  80  to  B  in  the  first  operation  ;  then  580  to  A  and 
7  to  B  in  the  second. 

The  process  here  explained  succeeds  in  separating  the  number 
into  the  terms  of  a  polynomial,  which  is  a  perfect  square.  Thus, 
344509  =  250000  +  86400  +  8169  =  (500  +  87)-^. 

2.  We  next  solve  the  same  problem,  abrldyirig  the  work 
as  much  as  j^ossible. 


roWKKS,    ROOTS,    AND    llADKALS.  liT  1 

Explanation.  —  The  miiuber  is  separated 
•A  4o  o9(  o87  juj^j  periods  of  two  figures  each  for  the  reason 
"*^  ..  given  above.     Having  found  the  first  figure  of 

j^jjlj  the  root  from  tlie  first  periocl,  it  is  square. I  ml 

11<57  )SUi{)  subtracted,  and  tlie  next  iH»riod  of  two  Hgures 

8101)  is  annexed  to  the  remainder.     Then   the    first 

figure  of  the  root  is  doubled  for  a  trial  divisor, 
anil  divided  into  045,  or  rather  1)4.  It  is  contained,  as  we  saw  above, 
not  9  but  8  times.  This  figure  8  is  written  afti-r  5  in  the  root,  and 
also  annexed  to  the  trial  divisor.  After  multiplying  by  8  and  sub- 
tracting, the  next  i>eriod  is  brought  down,  and  so  on. 

A  careful  comparison  of  this  solution  with  the  precetling  will 
make  the  whole  process  plain.  The  difference  is  that  in  the  second 
solution  all  the  ciphers  are  omitted. 

2.    Extract   the    square    mnf    ti\'    tin-    (Icciniiil     fi-action 

l-.\  ri.A.N  A  I  lo.N.  —  Here    as     bet'ore 
.00W02'70(  .01904  +      ,i„.  „,„„her  is  divided  off  into  periods 

^«  r^A/^  of  two  figures  each  commencing  at  the 

29  ).000202  ,     .       ,       .   ,      ,„,      -.    ,  -.  ,  ,, 

^        261  decimal  point.      Ihe  first  figure  of  the 

3804  UTCPOO  ^^^^^  having  been  found,  it  is  s<juar('(| 

152  10  and  the  result  subtracted,  after  which 

178400    etc.  tl><^  »*^xt  period  is  brought  down.    The 

first  figure  1  is  then  doubled  for  a  trial 

divisor  and  the  process  is  continued. 

A  cipher  is  annexed  to  7  to  make  its  period  full.     Thereafter  two 

new  ciphers  would  be  used  for  each  new  ])eriod.     The  second  trial 

divisor  :J8  not  being  contained  in   IT.  ;i  cipher  is  i)laced  in  the  root 

and  also  at  the  right  of  the  trial  divisnr.  after  which  a   new  period 

of  two  ciphers  is  annexed  to  tlic   rrniainder.     The  trial  divisor  is 

then  contained  4  times. 

271.  Rule  for  Extracting  the  Square  Root  of  Numbers  written 
in  the  Arabic  Notation. 

1.  ('onnueiH'in*;  at  tlw^  dccinuil  iM)iiit  divide  tlie  niimher 
otf  into  i>eriods  of  two  figures  eacdi. 

2  Find  the  greatest  square  in  tlic  left  hand  peri<xl,  ])lace 
the  root  at  the  right  subtract  and  bring  down  the  next 
period. 


272  TEXT-BOOK   OF    ALGEBHA. 

3.  Double  the  root  already  found  and  find  how  many 
times  it  is  contained  in  the  remainder  exclusive  of  the  right 
hand  figure,  and  place  the  quotient  in  the  root  and  at  the 
right  of  the  trial  divisor.  Multiply  by  the  figure  found, 
and  subtract,  after  which  proceed  as  before. 

a.  Fill  out  decimal  periods  with  ciphers  as  needed. 

b.  If  at  any  time  a  trial  divisor  is  not  contained  once  in  the  re- 
mainder place  a  cipher  in  the  root  and  one  at  the  right  of  the 
divisor. 

c.  Point  off  into  periods  commencing  at  the  decimal  point  (both 
ways  in  a  mixed  decimal),  and  place  the  decimal  point  in  the  root  at 
the  time  a  decimal  period  is  taken. 

272.   Exercise  in  Extracting  the  Square  Root  of  Numbers. 
1.   Find    the    square    root    of    2G01 ;    47089;    1772.41; 
433.4724. 


2.  V41.2164;  V965.9664  ;  Vl 7.338895  ;  V.()01()3041. 

3.  Extract  the  square  root  of  |  ;   '^'^j'l  ;    iV^^b  ;  US*. 

Kemai{k.  —  Sometimes  fractions  need  to  be  reduced  to  their  lowest 
terms  before  attempting  to  extract  the  root  indicated. 

4.  Find  to  within  less  than  .001  the  square  root  of  4.64  ; 
of  2;  of  9.999;  of  .00111. 

5.  Keduco  to  decimals  and  extract  the  square  root  of  the 
following  to  within  .00001 ;  i.e.,  to  five  places  :  — 

1  .   1  2  .   a  .      7 __ .   r^j  .   'in 

2  ?     -^;$  5     4  ?      IK!   ?     "7  f     "8* 


SECTIOX   III. 

Crui:  KooT  of  Poj^vxomiai.s. 

273.    Extraction  of   the  Cube  Root  of  Polynomials.     How- 
Effected. 

1.    Required  to  extract  the  cube  root  of  A^  -|-  3  A"-^  1>  -|- 


l'()\\i;i;s.    i;n..r<.     wi.    i;  \  i  »i('.\  i.s.  :i73 


A'  -h  3  A-li  +  '6  A13-  -h  J3 'C  A  +  li     Koot 
A8 


3A-  +  :;  AH  +  B^  )3  A2B  +  3  AB-'  4-  B" 
:;  A^B  +  ;}  AB-'  +  B=» 

Exi'LANATioN.  —The  first  term  of  the  root  having  been  obtained 
by  extract  in  ji;  the  cube  root  of  tlie  first  term  of  the  polynomial,  it  is 
eubed  and  the  resnlt  subtracted.  The  remainder  is  '4  \'^  B  +  3  A  B- 
+  B'. 

By  reference  to  258  we  know  tliat  A  +  B  is  the  cui)e  root  of  the 
polynomial,  and  that  the  second  term  of  the  root  is  B.  In'ow  we  see 
that  the  second  term  of  the  root  can  be  gotten  by  dividing  the  first 
term  of  the  remainder  by  three  times  the  square  of  the  first  term  of 
the  root  already  found,  or,  3  A'-^  B  by  3  A'^. 

If  to  the  3  A-,  which  becomes  a  trial  divisor  for  the  next  term  of 
the  root,  we  now  annex  3  A  B  +  B=^,  i.e.,  three  times  the  first  by  the 
second,  plus  the  s<|uare  of  the  second,  and  multiply  the  sum  by  the 
second  term  B,  ui)on  subtracting,  nothing  remains,  and  tlie  operation 
is  complete. 

It  <'an  Im'  show  n,  as  in  267,  that  the  binomial  solution  is  of  general 
ai>plication,  and  that  when  the  root  is  to  consist  of  three  or  more 
terms,  we  first  find  two  and  then  proceed  in  the  next  operation  as  if 
they  were  one  quantity,  and  so  on. 

2.  Extract  the  cube  root  of  S  ./•  -  :)(>  ,/• '  +  1 14  .r«  —  207  x='  +  L'ST)  .r^ 
-225j  +  12.-). 

t>jc-- ;w X" ■  +  11-1  .c  -*..  .,    ,  /  -  rjr) x  + 125  \'Zx--iix  +  i 

8*» 

12x<  —  18x3  +  «x»  |-»J«3  +  IHx* -'J07x 

I  -'•Wa;-'4-54x<-2ra<' 
12x»-36x«  +  27x» 


30ar»-46x 
-»-25 


36x1  +  57  x«-45x  +  26 


60  X*  -  180x3  +  285  x2  -  226  x  +  125 
60  x<  -  180  x»  +  286  x»  -  22ft  X  +  126 


Explanation.  —  The  12  x«  =  3  (2  x^)  2  ;  the  -  18  x»  =  3  x 
-3j;  X  2x2;  9^2  ^  (lix)^ 

In  tlie  next  operation,  12  x «  —  30  x«  +  27  x2  :::;  3  ( 2  x2  -  3  x  )2  ; 
30  X-  -  45  X  =  3  X  5  (2  x2  -  3  X) ;   25  =  52. 

274.    Rule  for  Extracting  the  Cube  Root  of  Polynomials. 

1.  Arrange  tlic  polyiioiniul  witli  ivterence  to  the  exixnuMits 
of  the  leading  letters,  extract  the  cube  root  of  the  first  term, 
and  subtract  its  cube  from  the  polynomial. 


274  TEXT-BOOK    OF    ALGEBKA. 

2.  Divide  the  first  term  of  the  remainder  by  3  times  the 
square  of  the  root  quantity,  and  the  quotient  will  be  the 
second  term  of  the  root.  ISlow  to  the  trial  divisor  is  added 
3  times  the  product  of  the  first  term  of  the  root  by  the  sec- 
ond, plus  the  square  of  the  second  (3  AB  +  B^),  thus  form- 
ing the  complete  divisor,  which  is  multiplied  by  the  second 
root  term,  and  the  product  subtracted  from  the  first  remainder. 

3.  If  this  does  not  finish  the  operation,  square  the  root 
quantity  (of  two  terms)  already  found,  and  multiply  by  3 
for  a  trial  divisor,  and  continue  as  before. 

275.  Exercise  in  Extracting  the  Cube  Root  of  Polynomials. 

1.  ^a^  —  maH^  54  ab''  —  27  P. 

2.  Sa^  —  S4.a^x-\r  294  ax'  —  343  x\ 

3.  x^-{-3x^-{-6x^-i-7x^-{-6x-'-{-3x-\-l. 

4.  a«  _  3  a'h  +  6  aW  —  7  a^b^  +  6  a'~  b^  —  3  ab''  +  b\ 

—  6  ahc. 

^3  Q  ^2 

6.  -,-'^+^xy-y\ 

y^       y 

7.  8  ic^  —  60  x^  z"  +  150  x^  %"  —  125  z^. 

8.  1  —  6  a;'"  +  12  £C  2;«  _  8  ^  3m_ 

9.  1  —  a?  to  three  terms  in  the  answer. 

10.  60  6'2  x"  +  48  cx^  —  27  c^  +  108  c^  a;  —  90  c*  iz;^  +  8  x'' 
—  80^3^:^ 

276.  Extraction  of  the  Cube  Root  of  Numbers  in  the  Arabic 
Notation  as  Polynomials. 

1.    Required  to  extract  the  cube  root  of  158252632.929. 

1.58000000  4-  2.52000  +  632.  4-.  929  |500  +  40 -|- 0  +  .9 
125000000 


750000 

60000 

1600 

33000000  +  252000  = 

33252000 

811600 

;:{:i4G4U0U 

3(540)2  ==  874800.00 

1458.00 

.81 

876258.81 

788000 
632 
788632  -f-  .929 

=  788632.929 

788632.929 

r<)\\Kl;>.     IKMds.    AM)    KADICAI.S.  Z<0 

Explanation'.  —  Starting  from  the  deciinal  point  the  niunher  is 
separated  into  parts  of  three  figures  eaeli  annexing  ciphers  to  fill  out 
omitted  orders.  It  is  plain  that  one  cipher  in  the  root  will  give  three 
in  the  power,  or  conversely,  three  in  the  power  will  give  one  in  the 
root,  six  in  the  ix>wer  will  give  two  in  the  root,  and  so  on.  Seeking 
the  greatest  cube  root  in  the  first  term,  oOO  is  found,  which  is  cubtnl 
and  subtracted  from  that  term.  Squaring  the  root  found  and  multi- 
plying by  :j  we  get  750000  as  a  trial  divisor.  Dividing,  the  quotient 
40  is  set  down  as  the  second  term  of  the  root.  As  in  the  polynomial 
rule,  taking  three  times  the  product  of  the  first  term  of  the  root  by 
the  second,  plus  the  square  of  the  second,  the  results  are  (>00(X)  an<l 
1000,  which  added  to  the  trial  divisor  make  HlltJOO,  the  complete 
divisor.  Multii)lying  by  the  second  term  of  the  root  and  subtracting, 
the  remainder  is  788000  to  which  the  next  term  is  added.  This  pro- 
cess is  now  repeated  for  the  third  term  of  the  root;  but  the  trial 
divisor  not  being  contained  in  the  remainder  a  cipher  is  placed  in 
the  root  and  tico  at  the  right  of  the  trial  divisor. 

2.    The  example  abridge;!. 

158'252',  032',  020'  (540.9    ' 
125 
7500  38252 

603 

10 

8110    (32404 
87480000  788(532029 
145800 


81 


87625881 


788032020. 


277.   Rule  for  Extracting  the  Cube  Root  of  Numbers. 

1.  Starting  at  the  decimal  point,  separate  the  yumber  into 
periods  of  three  figures  each. 

2.  Find  tlie  greatest  cul^e  in  the  left  hand  period,  place 
the  root  at  the  right,  subtract,  and  bring  down  tlic  next 
j)eriod. 

3.  Square  the  root  figure  found,  and  multiply  the  result 
by  3iM)  for  a  trial  divisor.  Find  how  many  times  it  is  con- 
tained in  the  dividend,  and  write  the  quotient  as  the  second 
figure  of  the  root. 


276  TEXT-BOOK    OF    ALGEBRA. 

4..  To  the  trial  divisor  add  30  times  the  product  of  the 
second  by  the  first  figure  of  the  root  and  the  square  of  the 
second  figure  for  tlie  complete  divisor.  Multiply  the  com- 
plete divisor  by  the  second  root  figure  and  subtract  tlie 
product. 

5,  Kegard  tlie  hgures  already  found  as  one  number  and 
proceed  as  before  to  find  the  third  or  remaining  figures. 

XoTE.  —  See  again  the  remarks  made  at  the  end  of  271. 

278.    Exercise  in  Extracting  the  Cube  Root  of  Numbers. 

1.  12107;  12812904;  1481.544;  107.284151. 

2.  .127263527;  .008741816;  ^^ 

2  <  44 

3.  Extract  tlie  root  to  third  decimal  of  517;  of  20.911  j 
of  5 ;  of.  2 ;  of  37. 

4.  Divide  the  cube  root  of    '^'^"'^^  by  the  square  root  of 

32768       "^ 

the  square  root  of  8.3521 


5-   y^;  S/2456;  ^999700.029999 ;  y  ^g^ 


^9791 
68921 


SECTION  IV. 

Extraction  of  Othek  Roots  of  Polynomials. 

279.   Other  Roots  Derived  by  Use  of  the  Square  and  Cube  Root 
Process. 

1.  The  fourth  root  of  a  polynomial  may  be  extracted  by 
taking  the  square  root  twice. 

2.  The  sixth  root  by  taking  in  turn  the  square  and  cube 
roots.     Either  operation  may  be  performed  first. 

3.  The  eighth  by  extracting  the  square  root  three  times. 

4.  The  ninth  by  extracting  the  cube  root  twice,  and  so  on. 


I'OWKliS,     Knors.    ANI>    l:  A  I  HCA  I.S.  Zii 

280.    Prime  Roots  other  than  the  Square  and  Cube  Roots. 

These  may  be  iouiid  by  means  of  rules  derived  ironi  inninda* 
in  the  same  way  that  the  rub's  inrs(juare  and  eub(  r^dt  wiir 
found . 

For  instance,  the  filtli  root. 

(A  +  B)*=  A^+(5A^-+  1()A»1J-H  l()A-r.--hr)Ar>»-f  B^)B 
Here  5  A*,  i.e.,  5  times  the  fourth  i>ower  of  the  first  term 
of  the  root,  is  the  ti-ial  divisor ;  and  to  it  must  be  added, 
for  the  complete  divisor,  10  A»B  +  10  A^B'-'  +  5  AB»  -f  B* ; 
i.e.,  10  times  the  cube  of  the  first  times  the  second,  etc.,  etc. 

281     Exercise  in   Extracting  Roots  other  than  the  Square 
and  Cube  Roots. 

1.  Find   the   fourtli    root   of   li'ya*  —  \Hi  c^x -\- 'J\i\  <i-.r' 

—  210  ax^  +  81  x\ 

2.  Find  the  fourtli  root  of  1  -  4  a:  +10  a-'  -  10  ./•»  +  V.)  .r' 

-16x'^  +  H)x^-4x'  -i-x\ 

3.  Find  the  fourth  root  of  625  x*  +  9()00  xY  -\-  4096  j/* 

-  10240  xi/  —  4000  a-V 

4.  Find  the  sixth  root  of  64  -  192a;  +  240 a--'—  KiO.r' 

+  60  X*  -  12  x^  4- a:^ 

5.  Find  the  sixth  root  of  x«  -  6  x»  +  15  x*  -  20  x«  4*  1  •">  ^•' 

-6X-I-1. 

6.  Find  the   eighth    root   of  x«  -|-  <S  a-'  -f-  2.S  x«  -\-  m  j'J> 

H-70x*-h56x»-f  28x^-|-Sx-h  1. 

7.  Find  the  seventh  root  of  128 x'  -  448  x«  -(-  (nli  a* 

_  r>m  r*  -f  2s(>  r^  _  84  y-  -f  14  r  —  1. 


278  TEXT-BOOK    OF   ALGEBRA. 


CHAPTER   XIX. 

OF    FRACTIONAL    EXPONENTS. 

282.  Fractional  Exponents  result  naturally  from  the  law 
of  exponents  in  the  extraction  of  roots.  By  129  each  ex- 
ponent IS  divided  by  the  index  of  the  root  to  be  extracted. 
When  therefore  this  index  is  not  contained  exactly  m  any 
exponent,  a  fractional  power  results. 

a.    To   illustrate.     (See   Ex.    266.)     y/7iFb^  =  a%^  =  a^h"^',    \/^  = 
'«"  =  c'';  etc.     If  now  instead  of  v^aW  we  had  Va'^^,  according  to 
the  same  rule  of  division  by  the  index  of  tlie  root  we  should  get 
Va'b^  =  a^b^. 

283.  Meaning    of   the    Terms    in  a    Fractional    Exponent 

The  numerator  of  any  fractional  exponent  evidently  denotes 
the  p)otver  to  which  the  quantity  is  to  be  raised  while  the 
denominator  indicates  the  root  to  be  taken. 

a.  The  expression  ^a«  means  the  ui'^  root  of  the  n'*  power  of  a. 
Hence  we  see  that  a  number  affected  with  a  fractional  exponent 
has  a  perfectly  definite  meaning,  and  performing  both  operations 
gives  rise  to  a  resulting  number,  the  value  of  the  expression. 

284.  Fundamental  Principle  Governing  the  use  of  Fractional 
Exponents, 

Any  quantity  affected  with  a  fractional  exponent  may 
be  separated  into  its  factors,  each  factor  taking  the  original 
fractional  exponent. 

This  principle  holds  for  integral  exponents,  thus,  (abe)"* 
=  aH^'c"". 

The  question  arises  does  it  hold  for  fractional  exponents 
as  well.     To  show  this  let  us  prove  that  the  expression 

m  m  m 

(1)   (ab)"  =a^'  X  b\ 


row  Kits,    uooTS.     \\i>    i;  A  I  lie  A  I, s.  279 

Kaising  botli  ineiubers  ot  (1^  to  the  n"'  power,  since  by 
definition  the  denominator  n  means  the  ?i'*  root  in  each  case, 

(2)  {ah)'"  =  a'"  X  h'"  =  aH""  (Ax.  o). 

The  resulting  equation  (L';  is  ])hiinly  true;  and  conse- 
quently the  members  of  equation  (1)  are  equal.  Moreover, 
it  is  easy  to  see  that  the  same  reasoning  would  apply  gener- 
ally to  any  fractional  exponents  and  to  any  number  of 
factors.  Hence  the  theorem  :  —  Any  root  of  a  product  is 
efjunl  to  the  product  of  the  like  roots  of  the  factors  :  and  con- 
versely, the  product  of  like  roots  of  two  or  inore  factors  is 
t'ljuiil  to  tJie  same  root  of  thru-  /nod net. 

(I.  If  a  ninnlwr  affected  with  a  fractional  exponent  is  to  ]n\ 
evaluated  it  is  immaterial  whttlit  r  it  )><■  t'nsi  r<ii.<i,l  to  the  jtonur  mid 
then  the  r(ntt  taken  or  vice  versa. 

This  amounts  to  proving,  for  example  in  «  ",  that  (a'")  "  =  [a  ")  «. 

We  have 
(f/w)  »  =  a  «  X  an  X  a  n    ...  to  m  factors.       (By  the  theorem  of  this 
article)  =  (a »)«.         Q.  E.  L.  (106,  8  (2.) 

As  a  particular  case,  (S*)*  =  (8^^  -  4. 

b.  A  quantity  with  a  fractional  czpaneut  Is  not  rhanuofl  in  value 
if  the  fraction  he  rcdured  to  other  terms. 

Thlls«^  =  u^  »A^  =  m^n'"^- 

For,  when  the  nunnTiitni  i>  .|niil»l.(l  ili.-  (|uantily  is  squared;  but 
when  the  denominator  is  doubh-d  a  s(juare  root  must  he  taken  in 
addition  to  the  root  denoted  by  the  old  index,  and  the  two  operations 
cancel  each  other.  We  may  reasqn  in  like  manner  for  any  other 
factor  by  which  both  terms  of  the  fractional  index  may  be  multiplied. 

285.  Fractional  Exponents  and  Radical  Signs  —  Kadical 
signs  are  used  for  the  same  jjurpose  as  fractional  exponents, 
and  the  two  notations  are  employed  interchangeably.  (See 
next  chapter.) 


280  TEXT-BOOK   OF    ALGEBIii^ 

It  would  he  for  in'oforahlr  [f  the  radical  sign  notation 
were  entirehj  displaced  by  fractional  exponents.  As  both  are 
in  constant  use,  both  liave  to  be  taught. 

286.  Object  of  Treatment  of  fractional  exponents  in  the 
present  cliapter.  —  Tliis  object  is  twofold  : 

1.  To  show  that  the  same  rules  apply  to  fractional  ex- 
ponents as  held  for  integral  exponents. 

2.  To  furnish  the  student  exercise  in  the  use  of  fractional 
exponents  in  all  the  simple  operations. 

287.  Fractional  Exponents  and  the  Fundamental  Operations. 

1.  Addition  and  subtra(;tion  of  quantities  involving  frac- 
tional exponents. 

Here  it  is  plain  that  we  can  always  add  or  subtract 
sitnilar  quantities  whether  affected  by  integral  or  fractional 
exponents. 

Tluis.  n  a¥  -f  ()  a¥  -  4  a¥  =  7  aW. 

2.  Multiplication  and  division  of  (piantities  involving 
fraction:d  ex})onents. 

Let  us  seek,  e.  g.,  the  i)ro(hu't  of  x''  by  .T^ 

x^  X  x^  =  x^  X  .r«  (284,  b) 

x^  X  x^  =  (.T^)«  X  (x')^'  (284,  a) 

(.r^  X  x')^  =  (.r«)«  ^  x^  (284) 

Now  §,  the  new  exponent,  is  the  sum  of  the  old  expo- 
nents -^  and  ^,  the  very  process  of  determining  it  giving 
their  sum.  In  other  words,  the  old  rule  of  adding  the  ex- 
ponents \w  multi])lication  holds  for  fractional  as  Avell  as  for 
integral    ex})onents. 

Since  division  is  the  reverse  of  niulti])licati()n,  the  ex- 
ponent of  tlic  divisor  is  stdifracfcd  from  tliat  of  tlie  dividend. 


I'oWKKS.     lands,     ANh    l:.\l)I("ALS.  281 

3.    Fractional  powers  and  roots  of  (piantities  affected  with 
fractional  exponents. 
Let  us  take,  e.  g.,  («')*. 

(«')'  -  {[(«')']'}*  =  {(«')'}♦=  {«•)» = {(«')'}♦  =  w^ 

=  «.\(284,a) 

since  the  fifth  root  of  tlie  tliird  root  is,  by  definition,  the 
fifteenth.  Now  ^  the  new  exponent  is  the  product  of  the 
old  exponents  f^  and  ^.  Hence  the  old  rule  of  multlphj'mg 
the  exponent  of  the  (quantity  by  the  exi)onent  of  the  power 
holds  ioT  fractional  as  well  as,  for  integral  exponents. 

Thus,  generally,  (a»)l  =  a'm  -^  and  conversely, 

flf,  IW     =   {(I  w )  «   =   (^<  v)    '»   =   (</  «)    «  =   (<t»»)  «  . 

a.  Since  fractional  exponents  are  governed  by  precisely  the  same 
rules  as  held  for  integral,  they  might  have  heen  used  from  the  begin- 
ning. There  are  weijjhty  reasons,  however,  for  deferriiii;  their  intro- 
duction until  now, 

288.    Exercise  in  the  Use  of  Fractional  Exponents. 

1.  .\dd  S^  4<>i,  and  1>7^ 

2.  W n i hi-  -f-  9 ^/ ^ hr'  —  1  ;'> ii i hi-. 

3.  15;//'y/*   -f"'"^"''  —  ni^  n^. 

4.  Find  the  sum  of  "1  a.v '"ij^ -\-\\  Uc — .\  "  ~  V  "  ^ -|- *► /> ; 
\\as-'"ip^'lhr-\-      "    .,  —  I'A:   ;in<l  "'''  —  A  +  T)       .,• 

5.  Find  the  sum  of  (> </  W/^  —\)c'^(l-\-\(),i'-^)A\  ^  (;  r^  i  />*  — 

„\f,l^(\rifl.    o^i,i  _y,„hi,i_:>„\f,i.     .,,,,1  _  'j„l/,i^,.^,l 

6.  From  tlH»  sum  of  5  ox^  —  (.r  -j-  //)*  -f-  (/r  —  h)^ ,  —  7  ^m"^ 
+  2  (j-  +  y)i  —  ^  (a  —  h)^ ,  and  12  ././-^  —  ;i  (.r  +  //)i  -f  12  (a 
—  //)*  take  the  sum  of  .'W/x*  -|-  4  (.r  -}-  y)i  -f-  ("^^  —  A)i  and  ^'.r« 

-(•'•  +  . '/'^  :!-•»(" -^:)*- 

7.  T(»  T)  .^r■-  —  7  f,r  -f  S  ;//i  —'Jr->'.  add  :'.  />/•  —  I  '•  "  + 
2  (iw'^.  and  then  sul)tract  |()  iir'  —  ."i  ./•//  -j-  .'J  ///•  —  12  r  "". 


282  TEXT-BOOK    OF    AIJJKiiUA. 

8.  From  9  c^ d"^ -{- 10  rj  d"^  —  17  chl^  t^ke  rJd^ —  20  c^d^ 
-  11  c'dK 

9.  Taking  the   even    roots    positively,  simplify  16^  +  8^ 
+  16^  +  125^  —  512^  +  100«-^  —  810-". 

10.  Calculate,  36 1%  49 ^^  4"=^%  8-^^,  9-<>-5 

11.  Multiply 

—  3  ?/n  and  6  y^ ;  also  9  a^ h'''  and  2a^h'^. 

12.  Multiply  together  7^ ,  7%  7l 

13.  a^'d'^-a'-a^^^l     a%^  '  a'^h  '  a-^]/^  =  2 

14.  a-^  •  .T~^  =  ?     m^  •  m^  =  ?     ?i  X  M^  X  ti^'  Xn  ~^  =  ? 

15.  Multiply  a^  -\-  a,-'^  -\-  a  hy  a-'^  —  (t^. 

16.  (^tH-'c^^-^)  -^(a^Z'-"c-i?^'^)=? 

17.  Multiply  2a-^  —  7x-h/—llrJ  by  r/a!?/-^^^. 

18.  Multiply  x'^-\-'if  by  ic^  —  yK 

19.  Multiply  m^  +  m^  ^^  +  ?i^  by  m^  —  nK 

20.  Multiply  a^  —  Z»-3  by  a^  —  b. 

21.  Simplify  (x^  X  a:  ^)'*. 

22.  Find  the  product  of  (^  ]Y^)  and  ( -^^ 

23.  Multiply  a?^  —  xy^  -\-x^y  —  y^  by  a*  -|-  x^  y^  -j-  y. 

24.  Multiply  a^  -\-l/^  -\-  a~^h\)j  ah~^  —  a^  +  h^. 

25.  Divide  11  a^  —  33  a^  by  11  (A 

26.  I )i vide  (^f  '5*  —  />   '^  by  a^  —  h~ ". 

27.  I  )ivi(le  x'^  -{-  a-^y^  -\-  y^  by  x^  ~\-  xhf  -{-  y^. 

28.  1  )i  vide  5  :^"  —  1 0  ^  ^ '"  +  15  x'^y  by  o  x^. 

29.  1  )i vide  x"^  —  a;^  —  .x-^  +  6  ^r  —  2  cc^  by  a;^  —  4  ic^  +  2. 

30.  Divide  a  —  h  ])y  <i^  —  /A 

31.  Square  the  following  : 


m     m 


_„^,^   _„M„.l    ^^  +  ^ 


(.-K  —  7/)' 


r(»\\i.i:s,    i;(.<(is.    .\NI>    i;aih('A1,>.  '2><i 

32.  (Jiibe  the  following:  a',  an  — a    nbn,  a  vh     p' 

33.  Simplify  ( -I  j  ^  (^„  i /,  i  ^-v^  (^,,  5/,Je--8;V 

34.  litiise  a'^  to  the  ^  power. 

35.  Square  a*  +  ^*  +  c*. 

36.  Extract  the  square  root  of  1  -|-  4  ir~*  —  2  ./•    '  — \  .r    ^ 
-\-25x    *  — 24  j--»  +  16a;-2 

37.  Factor  a*  —  1  ;  o-^  -  27 ;  a*^  -f"  ^'-'^  "h  ''''.  (135,  4) 
38          g  —  7  a;*        _t.  _  ^  _  _  ?       ^iLzL^  -_      ^f*      _  •> 

a;—  5ri— 14   *  a-*H-2       *        «/>  —  ^*       a^—b 


284  TEXT-BOOK  OF  ALGEBRA. 


CHAPTER   XX. 

RADICALS. 

289.  Radical  Quantities  or  simply  Radicals  are  quantities 
of  which  some  root  is  to  be  extracted.  They  are  expressed 
sometimes  with  radical  signs,  sometimes  with  fractional  ex- 
ponents.    (285.) 

a.  When  a  root  of  a  quantity  can  be  found  exactly  the  radical  sign 
or  fractional  exponent  disappears  in  the  simplified  form  (272).  It  is 
only  when  the  root  cannot  be  found  exactly  that  these  signs  have  to 
be  retained.  As  a  consequence  the  subject  of  radicals  deals  almost 
exclusively  with  the  reduction  and  use  of  quantities  whose  roots  can- 
not be  exactly  found. 

290.  Radicals  whose  Roots  cannot  be  exactly  extracted  are 
termed  Irrational  Quantities,  or  Surds,  (the  English  term). 

Thus,  ■\%  ^/7^  etc. 

a.  A  fraction  always  expresses  the  ratio  of  one  number  to  another, 
viz.,  that  of  the  numerator  to  the  denominator.  Upon  reduction  to 
the  decimal  form  it  gives  rise  to  either  a  finite  or  circulating  decimal 
{i.e.,  one  in  which,  sooner  or  later,  sets  of  figures  are  repeated.)  Now, 
the  process  of  extracting  the  roots  of  numbers  never  gives  rise  to  a 
circulating  or  repeating  decimal,  as  the  student  can  easily  see  with 
a  little  reflection.  Consequently  the  latter  can  never  exactly  equal  a 
fraction,  and  thus  express  a  rafio.  Hence  the  propriety  in  the  use 
of  the  word  In-ational. 

291.  The  Treatment  of  Radicals  embraces: 

1.  Kediu-tion  of  radical  quantities. 

2.  Addition,  subtraction,  multiplication,  and  division  of 
radical  quantities. 


I'OWKUS,     KooTS.    AND    KADU'ALS.  285 

3.    Eciuations  containing  radicals. 

a.  It  will  often  be  convenient  to  solve,  /»«»/<  /m.^.-m,  iIh-  f»aiii»-  t-wi- 
eise  in  tlie  two  notations.  To  indicate  this  brackets  will  be  written 
at  tlj»'  left. 

SECTION   I. —Reduction  of  Radicals. 

292.  Kinds  of  Reduction.  —  The  ti-catnient  will  U'  iiinler 
three  heads.  1.  Sini])litieati()n.  2.  liediiction  to  surd  form. 
3.    Reduction  to  a  coniinon  index. 

1.  -  SIMPLIFICATIONS. 

293.  Reduction  of  Radicals  to  equivalent  ones  having  a 
Lower  Index,  or  to  Rational  Quantities. 

1.  Simplify  (4f/2)4  ^  ^4^,,  (187.  /.) 

{  (4  ./•-)i  =  ((4  a')^  )*  ■=  (2  «)i  (287,  3,  and  129) 

(  \^4  «-^  =  y/  V4  a'  =  V2  a  (^et-  59.  // ) 

2.  Simplify  (1)  u*ir)^  =  -^i)  aV/^ 

(  (9  a*b'')^  =  ((9  «V/2)*  )*  =  Qi  a%y    (287.  :;.  and  129) 
(  y/^a*f^  =  y/  V9  a'b^  =  "v"?"^ 

3.  Simi)lify  (—  8  m«7t«)*  =  ^—8  vihi'^ 

^  (_  8  ?nV)*  =  ((-  8  w«««)i  )i  =  (—  2  //tn-')* 
',  -^'ZTHfn*?!^  =  y/  ^irir/w»7i«  =  V-  '2  nnr 

294.  Rule  for  reducing  Radicals  to  Lower  Indices.  —  Take  an 
exiK't  root  of  the  (pumtity  (whose  index  nmst  he  a  factor  of 
the  given  index).  The  result  is  still  affected  by  a  radical 
sign  with  the  other  factor  as  its  index. 

295.  Exercise  in  the  Simplification  of  Radicals  liy  nduction 
to  a  lower  index. 

1.  Simplify  -^Uya*b*c*y  y/21  a*d^^  ^/25<H) 

2.  Find  the  fourth  root  of  81  a\rh,\  \m y-,,\  ./,  .,y.-;^ 


286  TEXT-BOOK    OF    ALGEBRA. 

3.  Simplify 

4.  Simplify  "v^,  v^^.  Vix^i/fp'i,  Va''^  ''^iyy 

296.    Simplification  by  removing  a  Factor  from  the  Radical. 

This  reduction  depends  upon  284. 

1.    Simplify  (162)^  =  Vl62 

f  (162)^  =  (81)^  X  (2)^  =  i  9  X  (2)^    (284  and  270) 
1  V162  =  V8l  X  V2  =  i  9  V2  (52,  d) 


2.    Simplify  (25  a%y  =  V25  a«Z» 

r  (25  o^^^*)^  =  (25  «-)^  X  («/>)^  =  i  5  «  O'^O^ 
1  V25  a%  =  V25^-  X  -Vab  =  -j-  5  a  V^//> 


3.  Simplify  (24«>^3c^)^  =  V2r^^V 
(24  a^bh'-y^  =  (8  «,3^,3>>J  X  (3  ac')"^  =  2  .//.  (3  .^r^)^ 
^/24a*P?  =  ^8"^«  X  \/3^'  =  2  ./A  v:5  ./r- 

4.  Simplify  7  (625  a^b'c)^  =  7  -v^625  a'^b'c 

7  -^6257?^  =  7  a/(;25  ^^Z/'^  X  c  =  _[-  7  X  5  o^^'  a/7=  35  (^Z*  ^^ 


{ 


5.  Simplify  (3  ax^  =  V27  a^x^ 

^'WaFx^  =  V9  d^x^  X  V37t  =  i  3  ^/.r'^  V3  « 

6.  Simplify  ^^_ 


d 


^VTT  ~  ^^  V  "4"  ^  Vr/  -  ^  2/.  Vc 


297.    Rule. 

1.  Separate  tlie  radical  quantity  into  two  factors  one  of 
whicli  is  the  greatest  perfect  power  of  which  the  desired 
root  can  be  taken. 

2.  Exti'a;-t  th(^  root  of  tlie  fact(n' Avhicli  is  a  ])erfect  power 
and  place  the  result  in  tlie  coeiticient  of  tlie  other  factor 
affected  as  before. 


lN.\Vi;i;S,     !;(.(•!>.     AM)     I:.\1)1(AL8.  -^1 

298.    £xerci8e  in  removing  Factors  from  Radicals. 

1.    V288,  2  V75^  V  !-'"•'•.    \  -T8T 


2.    V9a*a;,   V^G  a»,  4  v-7t/^^S  V50aZ»-'c-,  V80a»;r» 


3.    S;i28  a'  ^»^  V—  108  x*  y»,  V25  (5  a»  +  5  a*  b)^ 

>^-512a«y»,  \^^iM-~^  5  \/80<y' 
4-    V72x*^  ^/96a'a:»,  V3179  a* ^»a-',  ^375 «*y^ 


5.    3  V32  «"  ^'  ^*,  5  (a  —  i)  Vca^^  f -h  ^  «^<- +  ^''<^> 

299.    Simplification  of  Fractions  under  the  Radical  Sign. 

«.  Tliere  are  good  reasons  for  i)rt*ferring  to  liave  (luantities  under 
the  radical  sign  intcf/ral.  This  object  can  always  be  attained  by 
means  of  the  well-known  principle  in  156.  In  order  to  remove  the 
denominator  from  under  the  radical  sign  both  tenns  of  tlu'  fraction 
are  nudti])lied  by  such  a  factor  as  will  make  the  denominator  a 
perfect  power  ;  tlu'n  extracting  its  root  the  result  is  placed  in  the 
d(>nominator  outside,  as  in  the  previous  case. 

1.    Siiuplifv  (5|)*  =  V'^ 

\2x'l/J       \2xhj     \.r.r)       \i^''y) 

=  . ±.(20  abxi/')^ 
2xy^ 

Hk.m.\uk.  —  The  student  may  write  this  example  in  the   radical 

notation.      He    should   accustomhimself   hereaft<'r  to    use    either  at 

pleasin-e. 

3.    Reduce  tii  f  "  J  -^m\l"  to  its  simplest  form 


»Y^  =  ,«,yV.  X-_  =  „,y   ,^  =  .,_^„,, 


'"  V  ''         .  r'' 


288  TEXT-B(J()K    OF    AJ.GEBltA. 

4.  Reduce  y  — '—  to  its  simplest  form. 

4  c-  // 

1(1^  X^  (IX         (1  (IX       jit^  1/  (IX 

Y  4<'"^  //       J  r  \  1/       2  (•  \  1/        1/       2  I'll       ^ 

..     ,.,.   'Ill  (  h'  \\ 

5.  .Sim])liiv  - ,-(         ,  ] 

S(>LUTI<>X. 

'Ill  (    h^  'lii\S        '111  h 

6.  Reduce  ( '-  |  =  V  '    to  its  simplest  form. 


ff  —  ./'  ! II  —  X  .  .  II  -\-  X  1 


\   II  +  X        V  II  +  X        II  4-  X        a  -\-  X 
7.    Reduce  y/v  ^^^  ^^^  sinii)le8t  forui. 

/TT        7         5  2"X3"X8        1    ,,— 


V  12  ""  V  2  X  2  X  3  X  2  X  3  X  3 


6 


300.  Rule  for  Simplifying  Radical  Quantities  by  removing 
the  denominator  from  under  the  radical  sign. 

1.  Simplify  the  radical  by  the  previous  cases. 

2.  Multiply  both  terms  of  the  fraction  under  the  radical 
sign  by  such  factors  as  will  make  the  denominator  a  per- 
fect power  of  the  degree  denoted  by  the  index  of  the  radi- 
cal, and  in  so  doing  use  no  unnecessary  factors. 

3.  Extract  the  root  of  the  denominator  and  place  it  as  a 
factor  of  the  denominator  of  the  coefficient  of  the  radical. 

301.  Exercise  in  Simplifying  Radicals  in  the  Fractional  Form. 

,        /I        /2     ,,    /3     1     /3     2     fn     flS\h 
'-   V/3'V5''V/8'3V7'5VT'(,25) 

9     aJ^^'     ^l2ll^'     (n'b'\h      p     la        ,e     lis 


POWKKS,     I:(M)'rs.     AM>     UADK'AI.S.  '2^0 


3a«ty/i^ 


II. -REDUCTION  TO  SURD  FORM       Converse  Operation. 

302.    Reduction  of  Entire  Quantities  to  the  Form  of  Surds. 

1.  Kediice  \\ax  to  the  form  ot  a  .s(nuire  root. 

W  or  =  \/9  rtV-*,  or  (9  a^^)  ,  by  squaring. 

2.  Change  —  \  (i%h  to  the  form  of  a  cube  root. 

3.  Put  —  3  a^x~'^  under  a  radical  whose  index  is  4. 

6   /o — 


«V5-;/¥=(^)' 


303.  Rule  for  Reducing  Rational  Quantities  to  the  form  of 
.surds,  or  ra(.lical.s  to  eciuivaU'iit  one.s  of  higher  degrees.  In- 
volve the  quantity  to  the  pro{)er  power  and  place  it  under 
the  sign. 

304.  Exercise  in  Reduction  to  the  Surd  Form. 

1.  Reduce  —  5  (I'b  to  the  form  of  a  cube  root. 

2.  Place  6,  2a%,  —  ;/,  —  7  m-ti  under  radicals  whose  in- 
dices are  2. 

3.  Reduce  J,  a  —  x,{j  a^x'*, '— —  to  the  form  of  cube  roots. 

_  ^ny 

4.  Express  Var*,  a^,  o  ,  and  a  as  sixth  roots. 


290  TEXT-BOOK    OF    ALGEBRA. 

5.  Express  V-,  2v<S,  ■>/(),  as    surds  of   the  same  order, 
viz.,  sixth  roots. 

6.  Express    Vo'   Vll'    vl^^  as   surds   having  the    same 
index. 

7.  Reduce  — to  the  form  of  a  cubic  surd. 

a  -{-  b  -\-  X 

V2 

8.  Reduce   ^  to  the  form  of  a  fourth  root. 

5 

305.    Reduction  of  Radical  Qualities  to  the  Form  of  Entire 
Surds :  In  Other  Words,  Placing  Coefficients  within  the  Sign. 

1.    Reduce  3V5  to  the  form  of  an  entire  surd. 


3V5  =  V3  X  3  Vo  =  V45. 
*    2.    Reduce  2  « V2  a^^  to  the  form  of  an  entire  surd. 

2  a^/2^  =  ^/J2~^'  X  V2^  =  ^{2ayx2a'  =  V64^ 

3.    Put  the  coefficient  of  —  ~--^  1/  — ^  within  the  sign. 

-l^M^-_JA^  x^  =  -v/^ 

^y'V3a2  Vl25/      3a'-^  V75/y^ 

306.  Rule  for  Inserting  Coefficients  under  the  Sign. 

1.  Raise  the  coefficient  to  the  power  denoted  by  the  index 
of  the  radical. 

2.  Multiply  this  result  into  the  quantity  already  affected 
by  the  sign,  and  write  the  product  under  the  sign. 

307.  Exercise  in  Placing  Coefficients  under  the  Sign. 

1-  "'  vn,  \  V3,  ~  y/^,  6  V4, 2  ^5,  i  (4)'  • 


3     ^4  /_f_      ^^     »  A^  —  ^"^  a  -}-  h  f  a  —  b\^ 


I'oWKI.s      l;(K»|s.     \\1>    I;AI)1(  ALS.  2'Jl 

4.  i'iace  tliu  -  111  Liiu  coetticieiit  oi  '2x-\/'.\(ib  within   tli«' 
sign. 

5.  (a  -  h)  y/a'  +  0-  +  L'  o/>.    /,\  H  a    3  x  V  liT 


III.  -  REDUCTION   OF  RADICALS  TO  THE  SAME  INDEX. 

308.    Reduction  of   Radicals  to  Equivalent  Ones  having  a 
Common  Index.     (See  284,  b.) 

1.  V3,  V^,  </¥)  =  (3)»,  (6)i,  (10)^. 

Evidently  a  common  index  can  be  found  by  reducing  the 
exponents  to  equivalent  oiks  haviuLC  ;i  common  denominator. 

(.S)4,  ((J)i,  (10)i=  (3)A,  (6)A,  (1())A  =  (3«)^  (6*)A  (103).\ 
or,       V6,  ^0,  -^lO  =  ^in  ^   ^il?  =  iJ/729,  ^1296, 

^looo. 

2.  Reduce  V«^,  v^a;^  V<?^*  to  surds  having  a  common 
index. 

(aa-;*,  (^a;2)i,  (ra--')i  =  (air)«,  (^»a:-^)«,  (car^)* 
«  (aV)*,  (^-^.r*)*,  (car*)* 
=  ^^^V.  v'/''.''.  v'o^. 

3.  Reduce  a  -y/jc  —  y/  and    ,  1  i  a  «'omni()ii  index. 

\  .'•  ^'  // 

'/   I  X     -   //)i,  /;  (./•  -\.  I,)-*  =  t,  (./•  —  //)«.  /,  (./•  -[•  If)-* 

4.  Reduce  a  -|-  c-  and  («  —  r)^  to  a  common  index. 

(a  -f  c),  (.^  ~  r)>  =  (r.  +  g)i,  (g  -  c)*    


292  TEXT-BOOK  OF  ALGEBRA. 

309.  Rule  for  Reducing  Radicals  to  a  Common  Index. 

1.  If   not  already  written  with  fractional    exponents 
place  them  in  that  form. 

2.  •  Reduce  these  exponents  to  equivalent  ones  having 
a  least  common  denominator. 

3.  Develop  the  radical  quantities  to  the  powers  denoted 
by  the  numerators. 

310.  Exercise  in  Reducing  Radicals  to  a  Common  Index. 

1.  ^/8,  V3,  ^6.  5.   3%  2%  oK 

2.  ^/a^,  Va.  6.    Va%  Va\  ^uK 

3.  a/5,  V4.  7.    </./',   \/.?';  ^.r^ 

4.  a^  and  bK  8.    a  V«  —  ^',  b  \a'^  —  x'\ 
9.  4  (5  x^y)"^,  3  (4  xi/)K  and  15  a  (3  Ox^)K 

10.  x^,  and  y« . 

11.  rt"*,  (8^/-;"".  (3^^ <•//}«. 

12.  (a  +  a')^,  (6t  —  ic)^ 

SECTION  II. 
Fundamental  Opekations  with  Hadicals. 

311.  Addition  and  Subtraction  of  Radicals.     (287,  1.) 

a.   Radical  quaUties  seendnyly  unlike  can  often  be  made  similar 
by  reducing  them  to  their  simplest  forms. 

1.  Add  together  3  V45,  V20,  and  7  Vo. 

3  V45  =  9  V5 ;   (296) :    V20  =  2  V5 ;    Adding   the 
coefficients,  we  have  18  Vo.     Ans. 

2.  Simplify  3  V28a'^  —  13  V252  a^x  -\- 15  a  V63  ax. 

3  V28  a^x  =  G  a  VT  ax ;  13  V2o2  ahi^  =  78  a  Vl  ax; 
15  a  V63  (Xic  =  45  a  Vfax-     Adding,  —  27  et  V7  ax-.     yl?is. 


POWKiis,  i:()()Ts,  AM)  i:ai)I(jals.  293 

3.  2V5+i  V«0  + Vl5'+2VI  =  ? 

2  y  I  =  §  Vl5^  (299)  ;    i   VOO  =   »  VFT;    2  Vf  = 
§  Vi5  Sum  »/ V15. 

4.  2  </i6^H- ^8i  -  a/:::5I2^  + -^192  -  7  </9r 

^Iy2"  =  4</3T  7%/ir=  7^3. 
Adding,  the  siun  is  found  to  be  12  a^. 

312.  Rule  for  Addition  and  Subtraction  of  Radicals. 

1.  Reduce  each  radical  to  its  sinijtlcst  ionii  by  one  of  the 
rules  for  simplification. 

2.  Add  the  coefficients  of  similar  tennh  as  in  sinii)le 
addition. 

«.  Similar  terms,  as  will  be  remembered,  are  those  which  have  the 
same  letters  affected  by  the  same  exponents  fractional  or  integral. 
In  the  language  of  radicals,  those  which  have  the  samp  index  and 
the  .same  quantity  within  the  sign. 

313.  Exercise  in  Addition  and  Subtraction  of  Radicals. 

1.  Add  8  vl2o  and  2  Vm-,    9  Vl92  and  7  V7r». 

2.  From  5  V30J"^y'take  3  V242  ay. 

3.  Add  10  a  V28^,  5  b  V63l^,  and  e  Vil2^ 

4.  vW4-5vT7?r— 2v^;  4  VT28  +  4  V7r>  — r>  Vf^. 

5.  2  V.3~-  I  VlT  4-  4  V27"  -  2  V7^ 

6.  V48^2  +  />  VWa  +  V3a(a  — 9  6)«. 

9-    V2rtar«  —  4  «ar  -h  2a  —  V2  ax^ -{- 4  ax -\- 2  a- 

/a  -f-  a; 

10.    From  (a  —  x)  Va*  —  ar*  take  r/  (a  -  x)  -\l- • 


294  TEXT-BOOK  OF  ALGEBRA. 

12.  3Vl47-^\/5-v/i- 

13.  i/W^b  +  ^/la%  +  a/o4  «^^  +  V^. 

14.  ^<S  r/^^*  +  16  a*  —  Vb'  +  2  aA^. 

15.  3  //-  (^^0^  +  -  (^^'^'')^  -  '•'  (  F  j  ■ 

16.  (54  a"'  +  6^>3)^  —  (16  a"' '  %')''  +  (2  «*'"  + ')  ^+  (2  c^a^O^ • 

17.  71  a/7^  +  4  ^0.21875  -  5  ^^0.0256. 

18.  c  V^^*WV^  -  a  VoiW?  +  />  -\/(7P?. 

314.    Multiplication  and  Division  of  Radicals.    (See  287,  2) 

a.  By  284  two  radical  quantities  can  be  multiplied  together  pro- 
vided they  have  the  same  index.  We  readily  see  that  they  cannot 
be  multiplied  until  reduced  to  the  same  index. 

Thus  V6  X  ^5  7^  V3()  i^  ^M. 

By  309  radicals  can  always  be  reduced  to  a  common  index,  and 
therefore  they  can  always  be  multiplied  or  divided  as  desired. 

1.    Multiply  V6  by  V3.     A7is.  Vi8  =  3  V2. 


2.  Divide  V^/*^'Vy  by  ^/a%f/\     Ans.  <Ja'^hx^. 

3.  Multiply  2  -v/3  by  3V2. 

2  (3)*  X  3  (2)^  =  2  (32)^  X  3  (2^)^  ^  6  (9  X  8)^  =  6  v'72. 

(287,2) 

4.  Divide  V2  a^x^  by  ^2  ax^. 

(2  a'x^f  =  (32  a^'^x^^y^ ;   (2  ax')^  =  (8  r/^ri^)^     Dividing,  we 
have  (4  ci'x^y^     Ans. 


POWKHS,    HOOTS,    AND    KADK  ALS.  295 

6.    Multiply  11  Vl^  —  4  Vl5  by  VO  -|-  VS. 
11  VL^-4  VB 

11  Vl2  -  4  V90  +  11  VlO  ~  4  V75, 
Or,  22  V3  -  12  VIO  +  11  VlO  —  20  V^  =  2  V3 
—  VTo     Atis. 

6.    I  )i vide  a  -\- 2  VaO  -\-  b  —  c  by  Va  -f  V^  +  Vc. 
a  +  2  V'7/7  -\-b-c  (Vo^-hVM^  V?. 
a  +  V<^  +  Vac  (Va  -|-  V?  —  Vc 

Va/>  —  Vat'  -f"  b 
Va3" +^>  +  V^ 

—  Vac  —  V^c  —  c 

—  Vac  —  V^c  —  c. 

315.  Rule  for  Multiplying  and  Dividing  Radicals. 

1.  If  the  radicals  have  the  same  index,  multiply  and  di- 
vide the  quantities  under  the  radical  signs  as  desired,  placing 
the  result  under  the  common  sign. 

2.  If  tlie  radicals  do  not  have  a  common  index,  reduce 
them  as  in  310. 

3.  When  polynomials  involving  radicals  are  to  be  multi- 
plied or  divided.  ])r()ceed  as  witli  rational  (piantities,  observ- 
iiiL:  the  rules  tm  the  iinilti])lication  and  division  of 
monomials. 

a.  The  roeflficirnts  arc  to  ho  lunhipliod  or  «livi(hMl  as  in<licatoc1  fo 
fonn  the  coefticu'iit  of  tlw  result. 

316.  Exercise  in  Multiplying  and  Dividing  Radicals. 

1.  2  Vr4  X  V2T;  3  V8  X  V6;  6  Va  X  2  V3^; 

^168  X  Vl47. 

2.  //  Vb*  X  b^  Va ;  V^  X  V| ;  2  V3  j-//  X  r>  V3^. 


296  TEXT-BOO\'    OF   ALG1-.15::A. 

3-   \/^  X  v/'l'"' ;  ^2-a  X  ^47^  5  V3  X  7  Vi  X  V2. 

4.  Divide  V40  by  V2 ;  6  V'54  --  3  V2;  77  a/9  --  7  -^. 

5.  Multiply  ^/lO  by  4  ^2;   V2  X  a/3  X  V^. 

6.  a  Vx  X  ^»  V^  ;  2  -^I  X  3  V|. 

7.  ^6ai^c-i  X  -v''3-i«-Hc'^;  ^ "^S  -^  ^  v"^. 

8.  a/2  •  -s/l'---  ^3 ;  (V5  +  2  V7  +  3  VlO)  2  V5. 

9.  Multiply  5  a*  by  3*;  AaH^  XoaH"^-,  ^S^e  X  i/~2ac. 

10.  Multiply  3a^^?/^    by  2x'^i/^  and  express  the   product 
without  fractional  exponents. 

11.  Divide  y^x^  by  ^/x^ ;   -y/y^  -j-  -\/?/^ 

12.  Multiply  together  j-  ^ax,  -y  VZ>//  and  —  -i/oz. 

13.  ( y/f  X  y/^)  ^  y  ^ ;  ^64  -  2 ;  V;^^^:^-  (./  -  a.). 

14.  (3H-V5)(3- V5);  (|+tV|)a-7V^.) 

15.  (^5-2^G)(3^4-^3(3);(y/^  +  y/|) 

16.  ( V2  +  V3  -  V5)  (V2  +  V3  +  V5) ; 

(,,2  _  ^  V^  _  5  ^,)  _^  (,,  _  3  V^). 

17.  («2_|_2aH^-4a^^»^  — 8^*^)  -f-  («^  — 4?>^). 

18.  ( V.^  +  Vv/4-  V.^) ( V^  —  Vy  —  V^) ( V^  —  Vy + v^). 

19.  (a/6  +  4  Vis  _  3  -  8  V2)  H-  V3  ;  (V7^  +  V32 

-  4)  --  a/8. 

20.  (2a/8  +  3  a/5  —  7  a/2)  (a/72  —  5  a/20  —  2  V2). 

21.  (2  V3  -  V2)  (2  +  a/9)  ;  (o  +  -^4  +  2  v^)  ( a/6  +  V5). 


i»oWi":i:s,  iiouis,  ANi)  i:ai>I(  Ai.s.  29' 


22.  Divide  20  by  V40;  m  \J  ~^  ^  '* y  ^TZl* 

23.  4  V2  X  (3  V8  -f-  i  Vli)  X  e  V2;  {a  +  ^  V^^) 

317.  Rationalization  of  Denominators.  —  There  is  one  case 
of  reduction  of  radicals  (292)  which  it  is  desirable  to  treat 
at  this  point.  It  is  that  of  rationalizing  denominators ;  i.e., 
transforming  fractions  with  radical  quantities  in  their  de- 
nominators into  equivalent  ones  whose  denominators  are 
rational.  Many  fractional  expressions  containing  radicals 
in  their  denominators  can  have  them  removed  by  multii)ly- 
ing  both  terms  of  the  fraction  l)y  the  ]>roj)er  factor  (156). 

2 

1.   Rationalize  the  denoiiiiiiaim  of 

V3 

2  2    ^  V3      2V3       .  .,.^, 

=  -      X  -  -  =    -71—     Am^.  (156) 

^/:^     v/3     V3        '^ 

* 

2.    Rationali/.*'  tl»e  (h^iioiiiiiiator  (»f 


2  +  V3 


4         _        4     _       2  —  V3       4  (2  —  V3) 


2  -I-  V3      2  -h  S/3  '    2  -  V3  4-3 

=  S  -  4  V3     Ans. 

3.    Rationalize  tin   dc  nniuiiuitor  of 
V3_4  V5  — 2  V7 
2  V3  -  V5  +  V7  * 
I^y  actual  multiplication  and  addition,  we  have 

V3-4V5-2V7__V3-4V5-2V7 
2  V3  —  V5^-f  V7  ^  (2  V3  —  V5)  +  V7 
2V3-V5~-V7      40  —  9  V15  -  5  V21  -f  6  V3g 
^(2V3-V5)-V7~  2(5-2Vl5) 


298  TEXT-BOOK   OF   ALGEBllA. 

To  rationalize  tliis  result  botli  terms  are  multiplied  by 
5  +  2  V  15,  giving  (the  student  should  verily  the  result), 

35  V~i5  +  35  V  1^— 70         2—  vI5— V^ 

2  X  —  6o  2 

K 


To  rationalize  a  fraction  of  the  ion 


a"'  ±  fj- 
luetjj  be  the  1.  c.  ni.  of  m  and  ?i.     Then 

.  .  .  =f  (f,i>Y  '  (?,y  130.  1  and  2) 

=  Q  (say) 

K 

If  now  both  terms  of      \  v   Ix'    inultiplieil    by    Q,   we 


have 


a'"  +  h 
K  KQ 


1         L_   —  /  ~i\p       7^'v'  since  the  (quotient  Q  multi- 

])lied  by  the  divisor  ^,*1  _[_^^^   ought  to  e<iual  tlie  dividend. 

But  (,,'^Y  and  (/^iY  or  ^J  and  ^/^  are  rational,  since  j^  was 

KQ 

taken    as  the   1.   e.  m.  of  7n  and  n.     Therefore  — is 

am         I)  >i 

the  equivalent  fraction  with  a  rational  denominator. 

In  Examples  2—4  the  multiplier  in  each  instance  is 
termed  the  eomplemenfary  radical.  In  the  case  where  the 
radical  or  radicals  are  square  roots,  as  in  Ex's  2  and  3,  wlien 
the  denominator  is  a  siirri  the  C()m])lementiiry  factor  is  a  dif- 
ference, and  vice  versa. 

318.    Rule  for  Rationalizing  the  Denominators  of  Fractions. 

1.  For  Monomial  Denonimators. — ^Multiply  both  terms 
of  the  fraction  by  such  a  factor  as  will  rationalize  the 
denominator. 


POWEUS,    UOOTS,    AND    liADHALS.  299 

2.  For  Biiioiiiiul  Deiiuiiiinators.  —  Multii>ly  both  terms  of 
the  fraction  by  the  factor  coniplcim'iitary  to  the  deiiomiiiiitor. 

3.  For  Polyiioiniiil  Dtii'iiiiiiiators.  —  Regard  one  pari  (  t 
the  denominator  as  one  t»'i m  and  the  others  as  the  second 
term  of  a  binomial,  and  pioc  ccd  as  in  2. 

a.  The  advantage  gained  by  rationalizing  denominatoi-s  may  be 

shown  by  an  example.     Thus,      ^      if  calculated  just  as  it  stands 

4  V? 
would  rc(iuire  the  extraction  of  two  roots  antl  the  division  of  one 
long  decimal  by  another.  Whereas  if  the  denominator  be  rational- 
ized, the  extraction  of  only  one  root  is  necessary,  and  the  division  is 
a  decimal  divided  by  an  intej^er  (usually  not  a  larirc  nninbor).  So, 
likewise,  with  binomial  denominators. 

319.  Exercise  in  Rationalizing  the  Denominators  of  Ex- 
pressed Divisions. 

m        2        _v3        :5+V8     a  -f  \/o  ^ 
^*     V^'   y/V)'   ^<r-'     L'  \  18   '    «</8^« 
SrcsoKSTiox.  —  Multiply  these  fractions  respectively  by 
V5,    \/3,    \/<>"-     \ -•     \-^^- 
3        6</2     -n/5      V^    5V4i-f3Vl2:6 


4. 


5. 


V3  +  V2 

'  Vio^ 

3V5- 
2  V5-- 

2  V2 
V18 ' 

1 

^'. 

V3- V2' 

V2- 

1' 

V7- 

V2 

V3- v/2' 

1  -f  2  V3 

VI  -f  ^' 

-VI- 

-r^ 

VI  4-^' 

.  8V$- 
*  5  va. 

-fVr: 
-  ()  V5 

:r?vs 

-^^' 

;[</; 

c  — 

+  ^ 

2ViT>-h8 
:.  -f  Vir> 

300                          TEXT- 

-BOOK 

OF   ALGEBRA. 

8.              1 

1 

1         1 

-\/x—    -yjy\x 

+  V.x^ 

—  1         X  —  Vx"  - 

•1 

9,     V2  +  A/3. 

V2  -  ^3 

10.    1  +  V3  +  V6 

5      

V2 

1  -f  V2  -  V3       V2  +  V3  -  V5 

320.  Powers  and  Roots  of  Radicals.  For  Principles  and 
Rules  see  287,  3. 

a.  Cases  arise,  however,  in  the  extraction  of  roots  for  which  the 
rules  heretofore  given  do  not  apply.  These  problems  will  be  treated 
separately  in  the  next  article. 

1.  Square  3  ^^3. 

(3  -^3)2  =  (3  iZff  =  9  (3)'  =  9  (3'^)^  =  9  a/9. 
Or,     (3  ^3)2  =  3  ^3  X  3  ^3  =  9  v'9.  (315) 

Cube  -s/ax^ ;  raise  ^  —  10  to  the  fourth  power. 

2.  Raise  .\yO  to  the  fourth  power;  2V;V  to  the  fifth 
power. 


3.    Raise   |V  —  ^xhj  to  the  fifth  power;  —  3  Vf  to  the 
third  power. 

^•(-^-)^v/(i)V(iT■(v/^" 

5.  (^^'^yf,  (v'2)2;  raise  V27^  to  the  nW\  power. 

6.  Square  V3  +  ic  V3 ;  cube  2  -  V3. 

7.  Square    a-'  —  ?/-§;    raise  V3  —  V2    to    the    fourth 
power. 

8.  Raise  V^c  —  2/  to  the  third  power ;  cube 


-\/  a  -{/a  -f-  x 
9.    Cube  V^  +  3  Vy;  cube  —  -^'^a  —  ^bc. 


POWERS,    HOOTS,    AND    KAMI  ALS.  801 

10.    Extract  the  square  root  of  ^a*;  ol  (Va^)^- 


11.  Exti-act  the  fourth  root  of  IG  a^  ^/2  c  j  of  y/y/a*b'^ ' 

12.  Extract  the  cube  i-oot  of  32  -J^27  a*x^y  of  (\/49~i5^)*  • 

13.  ^6T««~S^^  Vy^,  \/l25"^7^y/3     . 

14.  Required  the  cube  vuot  of  125  a:  i,  of  04  a^l>*  V2  cd. 

15.  Extnict  the  fifth  root  of  486  a  -^ia^. 

16.  Extract  the  scjuare  root  of  a-J -f  Ga:§//i -f  <> //;  tlie 
cube  root  of  (a  -{-  x)  y/a  -\-  x. 

17.  Exti*act  the  fifth  root  \/32  x" ;  the  eighteenth  root 
of  \/^»«^'-». 

18.  Extract  the  cube  root  of  8  Vx*  -\-  3(>  xf/  -f  54  i/^  Vx 

+  27y^. 

19.  Extract  the  square  root  of  9  x  —  6  Vary  -f  //  —  ^>  Vx 
+  2  V^  +  1. 

321.  Roots  of  Radical  Quantities  not  in  the  Original  Pro- 
duct form.  —  Binomial  Surds. 

a.  If  X  and  y  be  replaced  by  2  and  3  respectively  m  Ex.  18  of  the 
last  article,  we  have  16  V^  +  '^^^  x  2  x  :]  +  54  x  9  V^  +  ^7  X  27  = 
945  +  502  V*^  which  has  only  two  terms,  one  being  a  surd,  and  may 
therefore  be  appropriately  called  a  binomial  surd.  In  order  to  extract 
the  cube  root  of  945  -I-  502  V^  by  the  usual  cube  root  rule,  it  be- 
comes necessary  to  arrange  it  in  the  form  first  given.  This  is  gen- 
erally difficult  to  do,  when  one  does  not  know  how  it  was  obtained. 
Such  roots,  when  they  exist,  may  often  be  discovered  by  special 
processes. 

Only  the  sim])lest  case,  viz.,  the  square  root  of  binomial 
surds,  is  presented  in  most  treatises  on  elementary  algebra. 
The  method  of  solution  for  other  cases  is  very  similar.  Two 
or  three  lemmas  are  a  necessary  preliminary  to  the  deriva- 
tion of  the  root  formula. 


■\/'H 

=  «  + 

V 

m 

n 

=  a^  + 

2 

a  -\Jm  -\-  m 

_  2 

a  ^vi  ■■ 

= 

a/'  -\-  rn  —  n 

s/vi  - 

= 

a-  -\-  hi  — 

n 

2  a 

302  TEXT-BOOK  OF  ALGEBRA. 

222.    Theorems  Relating  to  Equations  containing  Radicals. 

1.  The  square  root  of  a  rational  qua7itity  cannot  he  partly 
rational  and  'partly  surd. 

Let  n  be  any  number,  and  suppose  it  has  for  its  square 
root  a  +  -yjni .     Then 


(Ax. 


Thus  a  surd  equals  a,  rational  fraction  or  a  whole  number 
(dei)endent  on  whether  2  a  is  contained  in  the  numerator 
exactly)  which  is  impossible.     (See  290,  a.)     Q.  E.  D. 

2.  In  any  equation  containing  radicals,  the  rational  part 
on  one  side  equals  the  rational  part  on  the  other,  and  the 
surd  part  on  one  side  equals  the  surd  part  on  the  other. 

Thus  in  the  equation  a  -f-  V^  =  x  -\-  Vy,  if  ci  represent 
all  the  rational  quantities  on  the  left  side,  and  x  those  on 
the  right  side,  they  must  be  equal ;  also  V^  =  Vy. 

«+V^=.T  +  V//      _  (Hyp.) 

V/>  =  a;  —  a  +  -yj'y  (Ax.  2) 

Now,  if  X  is  not  equal  to  a,  -yj  h  is  partly  rational  and 
partly  surd,  which  is  impossible  by  1  of  this  article. 
Whence  x.  must  equal  a,  and  if  x  =  a,  V^  =  V?/-    "(?•  J^-  ^• 


If    Vrt  4-  V^  =  V.r  +  V^/,   then    will   ^ a  -_^h  = 

Vx  —  Vy 

V^+  V^  =  V^  +  y^  (Hyp.) 

aJ^^f,=x^2  Vxy  +  y  (Ax.  5) 

^.^  a  =  X  -\-  y,  and  V0  =  2  Vxy  =  V4  xy        (2,  above) 
.'.  a  —  -yjb  ^x  —  2  ^xy  -\-  y  =  {^x  —  ^yf 
...  V(i- V^  ^  Vx  -Wy.     Q.  E.  D. 


POWERS.    ROOTS,    AND    RADICALS.  803 

323.  The  Square  Root  of  Binomial  Surds.  —  Demonstration, 
Rule  and  Examples. 

1.  Let  A  represent  the  rational  part  and  VB  the  mdical 
pai-t  of  any  binomial  surd.  ^Suppose  Var  +  Vy  to  represent 
its  square  root.     Then 


(1)  Va  -f  VB  =  V  a-  -f  Vy 

(2)  A  =  X  -f  //  and  (3)   B  =  4  xy  (322,  3) 
Now,  {x  —  i/Y  =  (x  ^-yY  —  ^xy^  A-^  —  B     (Identities) 


(4)  .-.  x-y  =  ^^:'-\^ 

(10          .r  +  y  =  A. 

2x      -A  +  VA'^-B 

(Ax.  6) 
(Ax.  1) 

,-_i/A  +  VA«-B 
^x      \l              2 

2//=      A-VA^'-B 

(Ax.  (!) 
(Ax.  2) 

,__./A-VA-^-B 
Vy  -  V            2 

Hence, 

(A.i.  (i) 

Also, 

H  +  ^A- 

-VA^ 
2 

-B 

Vi  -  V^  =  y/A  +  VA^-B_^A~VA-^- 


B 

2 

(322,  3) 


2.  RrLE.  —  Tlie  rule  is  written  from  the  foi-mula  just 
found. 

(1.  Sc^uare  the  rational  quantity,  subtract  the  quantity 
under  the  radical  sign  fnun  the  result,  and  extract  the 
square  root  of  the  remainder.  If  this  cannot  be  done 
there  is  no  root  of  the  form  sought. 

(2.  Add  the  root  just  obtained  to  the  mtiontvl  quantity 
and  also  subti'act  it  from  the  rational  q\iantity,  aiid  divide 


304  TEXT-BOOK   OF   ALGEBllA. 

the  results  by  two.  The  sum  or  dittereiice  (according  as  the 
binomial  surd  is  a  sum  or  difference)  of  the  square  roots  of 
these  quotients  is  the  root  sought. 

((.    Instead  of  following  the  rule,  infipectioii  will  often  enable  one 
to  determine  the  root. 

3.   Examples. 

(1.   Extract  the  square  root  of  31  +  10  V6. 
Here  A  =  31,  and  VB  =  10  VO  =  V600,  or  B  =  600. 

A2  -  B  =  961  -  600  =  361 ;  and  VA^  -  B  =  19 


/- 7-         ,  /  31  +  19    ,   ,  /31  -  19 

.-.  V31  +  V600  =  y — |—  H-  y  ■ — 2 —  ^ 

_^_  (5  _|-  V6)     Ans. 


2.    Eeduce  V '/ijo  +  2  m^  —  2  m  Vnjj  +  ^f^^  to  its  simplest 
form. 


Here    VA^  —  B  =  \^n'^p'^  =  /ip.     Hence,  the   answer  is 


3.   Vll  +  2  V30  =  ?    Vll  -j-  2  V30  =  V6  +  2  V30  -|-  5 
=  V6  -f  V5     Ans. 

324.   Exercise  in  Extracting  the  Square  Roots  of  Binomial 
Surds. 

1.  3  -  2  V2,  49  -  20  V 6,   87  -  12  V42,  10  -  V96; 
42  +  3  Vi742. 

2.  75  +  12  V2T,  4^  -  ^  V3;  I  +  V2: 


3.  £c  —  2  Va;—  1,    a;  +  ic?/  —  2  ic  Vy,    2  —  V4  —  4  a'^ 

aaj  —  2  a  V«^ic  —  a^,  X  -\-  y  -\-  z  -f-2  V^c^  +  y^. 

4.  Solve  by  inspection  4  +  2  V3,  6  -  2  V5,  9  -  2  Vli, 
23-8  V7,  11  +  V72,  28  -~  5  Vl2. 


roWEliS,    KUUTS,    AM>    KADiCAL6.  305 

5.   Extract  the  fourth   root  ot  17  +  12  V-?,  50  -f-  24  V^) 
^  V5  4-  3i,  248  +  32  VOO. 


6.  Va  V5  +  V40  ==  ?     V  V1573  -  4  V78,  V2]  +  V5. 
y  ^  -h  [;  Va'  -  cS  V28  +  10  V3  +  Vg7  -  1()  V'S. 


325.    Fundamental  Operations  with  Imaginary  Radical  Quan- 
tities. 


Any  pure  imaginary  quantity,  as  v  —  a,  can  be  reduced 
to  -^  Va~^V—  1  (284),  of  which  Va  may  l>e  regarded  as 
the  coefficient  of  the  imaginary  V—  1.  Thus,  all  pure  im- 
aginaries  can  be  expressed  in  terms  of  V—  1.  Let  us 
examine  the  powers  of  V—  1. 

(  V—  1  )^  =  -  1  by  dejinltion  of  a  square  root. 

( v^^i  y  =  ( V- 1  y  V- 1  =  - 1  v^=T 

( V  - 1  )*  =  ( v^^i  y  ( v=n  )2  =  - 1 X  - 1  =  + 1 

(  V—  1  y"  =  ((  V—  1  )*)"  =  +  1,  where  71  is  any  integral 
number. 


( V--T)*"+i  =  ( V- 1)*"  V-^l  =  V-  1 
( V^^)*"-^*  =  (  V-'i)*"  (V— T)2  =  -  1 


1.    Add  V—  a'^  and  V—b'K 


.-.  V— 7a''  4-  V—b^=  (  ±  «  i  6)  V— 1.     Or,  taking  posi- 
tive roots  only,  =       (a  -\-  b)  V—  1. 


2.   From  4  V--27  take  2  V—  12 


4V--27  =  12>A-3;  2V— 12=4V-3; 


...  4  V--27  -  2  V-  12  =  8  V—  3. 


306  tp:xt-book  of  algebha. 


3.    Multiply   c  -\/  —  ahy  d  ^  —  b 


c  V  —  a  =  c  -\  a  ^ —  1  ;  d  ^ —  h  =  d  -sjb  V —  1  ; 
now  since  V—  1  X  V—  1  =  —  1,  -l^y  delinition, 
c  V—  (t  X  d  V —  b  =  cd  ^/ab  X  —  ^  =  —  c  d  ^ab. 

Note.  —  It  should  be  remembered  that  the  sign  ±  is  only  written 
in  case  of  ambiguity  (See  264).  When  it  is  known  by  what  factors  a 
product  has  been  formed  the  appropriate  sign  must  be  prefixed. 

Thus,  V+1  X  ^/^^l  =  +  1  ;  V^  x  v'^  =  - 1 

Here  c  V—  a  X  d  V—  b  equals  not  ±  cd  V-\-  ab  but  —  rd  v  +  ab. 

4.    Divide  3  V^^  —  2  V— 11^  +  VO  —  9  by  11  V -^. 

(3  X  2  V^^  —  4  V:>  V^^  +  V<J  -  1))  -^  3  V2  v^^ 

.     ,        V3 

=  V2  -  i  V^'  + 

•'       -'       3  V-1        V2\/-l 

8ini])lit'yini;-  tliesc  ttM-ms,  we  have 

,-       o     .-        V3  V3  V^^- 

_V-=^.  3 3 V^n     V2 

~     -^ "  v2  V-  1  ""  V2  V^=^  ^  V-H" "  V2 

3  V-^ 


/-    ,        V3  3 

""^^^'^'-^^^  +  3V^-V2V-T 

V~  3  

-  V2  —  ■-;  VO  —  — ry — •  +  %  V—  2     J/iS. 


5.    Square  (V—  a  +  V—  b) 


POWERS,    KOOTS,    AM)    i:AIH('ALS.  o07 

6.    Extiuct  the  sqiiun*  root  oi  4  V —  G  —  2. 

Kvidently  the  ])i'odiict  midr;-  ilic  nulic-al  sij^ii  must  be 
—  -4  (since  2V — ()  =  \— IM).  This  numl)er  suggests 
the  root  V—  0  4-  V4-4  =  V—  (>  +  2.  Squaring  V—  (> 
+  2,  we  get  —  ()  4-  4  V^^  +  4=4  V^^^  —  2. 

326.  Exercise  in  the  Application  of  the  Fundamental  Rules 
to  Imaginaries. 


1.  Add  2  V—  48,  ;{  V—  12,  5  V—  .S,  and  —  7  V-  o2. 

2.  Find  the  sum  of  2  +  V—  1  and  3  —  V—  04. 

3.  Add  -v/^=^  and  ^^TTi ;  -J/Hl  and  ^/^9. 

4.  From  V —  18  subtract  V —  H;  from  o  -\-  V—  ^  take 

a  -f-  V—  c. 

5.  Multiply  4  V- 3  by  2  V-  2  ;  4  V^^  X  9  V-~12. 

6.  2  V^l>  X  r>  V-~4  X  3  V=^;  2  ^/::^T  x  :;  v  -  To. 

7.  Find  the  third  and  fourth  jn^wers  of  aV —  1. 

8.  I  )i  vide  (>  V^^^  by  2  V^=^;  (4  +  V^-2)  -4-  2  —  V—  2. 

9.  Divide  2  \/S—  \^—  10  by  —  V— ~2";  —  V^^^-^  — 
(>  V-  3). 

.      ,        1  +  V—  1  S 

10.  Simi.lilv     -  ; 

'     '   1  _  V—  1     -  1  -f  \  -  u  ;; 

11.  (7      \  -o)  (10—;;  \  -<;,:  <  .:■  _  T)  v^"  V  — T)  (7 
_  I  \  :;  \  _T). 

12.  Of  wliat  number  are  24  +  7  V—  1  and  21  —  7  V—  1 
tht»  factors  ? 

13.  Find  tin-  c-Dntinucd  ]>rndiict   i^\'  ./•  -f  ".  ./•  -f-  "  V —  1, 
«  —  a,  and  x  —  a  V—l* 

14.  Multiply,  (x  —J,  —  y  V—  1)  (x  —p  +  y  V— T). 
16.    Kaise  a  -j-  It  \^—  1  to  the  third  power. 


308  TEXT-BOOK   OF    ALGEBRA. 


16.  Extract  the  square  root  of  o  -j-  ^  V—  22. 

17.  Extract  the  square  root  ofSV— 1  (=0  +  8  V—  1. 


18.    Extract  the  square  root  of  —  2  +  4  V—  6  ;  of  31  +  42 

19.  ( v^nrr  +  V^==l9)  (V-iT9  -  V-T33)  =  ? 

20.  Prove  that  an  imaginary  quantity  of  the  form  V—  a 
cannot  be  partly  real  and  partly  imaginary,  as  V —  a=  m 
_|.  ^ZZVi,     (See  322, 1.) 

21.  Prove  that  in  any  equation  containing  imaginaries  of 
the  form  a-\-h  V —  1  =  m  +  ^  V —  1,  the  real  part  on  one 
side  equals  the  real  part  on  the  other,  and  the  imaginary 
part  on  the  one  side,  the  imaginary  part  on  the  other.  In 
particular,  if  a  -\-h  V —  1=0,  then  a  =  0,  and  ^  =  0. 
(See  322,  2.) 

Remark  on  Imaginaries.  —  It  was  not  till  the  beginning  of 
this  century  that  Argand  invented  his  diagram,  since  which 
time  great  progress  has  been  made  in  this  and  allied  sub- 
jects. An  extended  presentation  of  imaginaries  would  be 
out  of  place  in  any  except  an  advanced  algebra ;  but  some 
little  idea  of  the  graphical  representation  of  imaginaries  will 
probably  be  very  suggestive  and  instructive  even  here. 

It  was  early  shown  that  multiplying  a  positive  number  by 
—  1  reverses  its  character  and  makes  it  negative,  and  mul- 
tiplying a  negative  number  by  —  1  reverses  its  character 
and  makes  it  positive.  Naw  we  may  conceive  of  —  1  as  an 
operator  which  revolves  the  whole  positive  series  around 
into  the  negative,  and  the  negative  around  into  the  positive, 
zero  being  the  pivot. 

But  we  saw  that  V— 1  X  V—  1  =  —  1,  or  V—  1  X  V—  1 
X  a  =  —  a;  i.e.  the  operator  V —  acting  on  a  twice  revolves 
it  from  the  positive  to  the  negative  position.  We  are  thus 
led  to  the  idea  that  V—  1  acting  on  a  once  would  revolve  it 


POWERS,   nOOTS,   AND  RADICALS. 


309 


half  way,  or  into  the  position  at  right  angles.  Generalizing 
this  result  we  have  the  iniaginaries  as  represented  in  the 
figure. 


.fiovri 

+  »-/31 

+  »V=1 

+  Tv^ 

+  tv=i 

-t.«vcn^.~ 

.^« 

■*-*V=i 

! 
5 

+  *V=1 

: 

a. 



+  2>/^ 

i 

i 

+     ^ri 

10,-». 

-»,-:,  -6,  -  3.  -  f.  -a,  - 1',  - 

i   0 

,-»-4, +B,+6,+r, +8. +». +10, 
f 

« I... 

A    1 — r 

1 

0  *      

—  4^—1 

..J4. 

In  Chapter  IT.,  we  regarded  tin*  luimhors  of  the  double 
series  as  fixing  lengths  and  directions  on  the  line  containing 
it.  Thus,  0  fixed  tlie  origin,  -f-  3,  the  distance  to  the  point 
3  units  to  the  right  from  the  origin,  and  so  on.  Similarly, 
here,  -|-  2  V—  1  fixes  the  line  from  the  origin  to  the  point  2 
units  above  it,  and  —  4  V—  1,  the  distance  from  the  origin 
to  the  jwint  4  units  below  it.  In  Argand's  diagram  the  dis- 
tance and  direction  of  any  point  in  the  whole  plane  may  l>e 
fixed  by  a  complex  number^  i.  e.,  one  partly  real  and  partly 
imaginary,  as  ::i:  a  ^  ft  V—  1,  by  letting  a  denote  the  per- 


310  TEXT-BOOK    OF    ALGEliKA. 

pendicular  distance  to  the  vertical  line  (measured  parallel 
to  the  real  series),  and  h  the  perpendicular  distance  to  the 
horizontal  line  (measured  parallel  to  the  imaginary  series). 
Thus,  the  line  joining  0  and  1  in  the  figure  would  be  repre- 
sented by  4  -|-  5  V—  1 ;  the  line  joining  0  and  2  by  —  3  + 
2  V—  1 ;  the  line  joining  0  and  3  by  — 4— 4V-"1;  the 
line  joining  0  and  4  by  +6  —  5V—  1. 

The  meaning  of  sums,  differences,  products,  and  quotients 
of  complex  numbers  may  now  be  interpreted  on  the  diagram. 
But  the  examination  of  these  operations  would  lead  us  too 
far  astray  from  the  subject  of  radicals. 


SECTION   III. 

Equations  Involving  Radicals. 

327.  Equations  containing  Radicals.  —  Since  there  is  no 
reason  why  such  equations  should  be  of  one  degree  rather 
than  another,  this  subject  might  be  postponed  until  the 
solution  of  quadratic  equations  is  learned.  Still,  because 
the  preliminary  transformations  are  much  the  same  in  all 
cases,  they  can  be  treated  here.  Only  such  as  reduce  to 
simple  equations  are  given  in  the  following  set. 


1.    Solve  Vic  +  5  +  ;3  =  8  —  Va; 
V^-f5  +  3  =  8  —  V^ 


^x  -^^=-5  —  Wx 

(Ax.  2) 

x-\-5=2o  —  iO  V^  +  X 

(Ax. 

5,  114,  2) 

10  -Vx  =  20 

(Ax.  a) 

Vx  =  2  • 

(Ax.  4) 

x  =  4. 

(Ax.  5) 

Solve  2i/5x-35  =  5^2x-7 

S(5x-35)  =  125  (2  0^  -  7) 

(Ax.  5) 

x  =  2^ 

(219,  2) 

»(>WEKS,    UOOTS,   AND   UADICALS.  311 


3.    Solve  \/12x  — o-l- V3ar  -  1  =  V27x  — 2 

12  r  —  5  +  2  y/Mx^  -.21x  +  f>  +  .'5  x  -  1 

=  27 x--_2__  (Ax.  5,  114,  1) 

V'3(>  X*  —  21  X  -\-  iS  =  iSx  -\-  2  (Axs.  a  and  4) 

3(>  X-  —  27  a-  -h  5  =  3()  j--^  -|-  24  :r  +  4  (Ax.  T)) 

.-.   ./•  =  ,V     ^1/^s-.  (219,  2) 

But  upon  substitutinj^  this  value  in  the  given  equation  we 

find  it  is  not  verified. 


Thus,  V-^W  +  V-  n  7^  i  V-  H 


i.e.,  9  V-  Vt  -f  4  V-  iV  ^  i  5  V-  ^ 
It  is  sufficient  for  our  present  purposes  to  know  that  values 
of  the  unknown  obtained  by  such  ])rocesses  as  these  may 
not  satisfy  the  original  equatin,  and  are  therefore  not  roots 
of  it  unless  the  equation  is  taken  in  a  certain  way.  In  the 
present  case  ^^  is  a  root  if  the  sign  of  the  first  or  second 
terms  be  taken  negatively. 

To  avoid  mistakes,  roots  {)l)tained  for  radical  ecpuitions 
should  be  verijied  by  substitution,  in  the  original  form  of  the 
equation.     (See  360.) 

4.    Given  a;  +  «  =  v  ^^  _j_  x  V^*  +  x^  to  find  x 

x2  -f-  2  flj;  +  a^  =  a""  +  x  V^^T^'^  (Ax.  5) 

x^  -\-  2  ((X  =  X  -s/li'  -f-  ./••-  (Ax.  2) 

X  -I-  2  tt  =  y/b'  +  X'  (Ax.  4) 

a.2  ^  4  ax  H-  4  rt2  =  ^2  4.  x'  (Ax.  5) 

//-'  -  4  «2 
.•.    .r  =    - . 

4  a 
Verification. 

6*       ,  /  .  ,  6«-4a»    /l6o2(>2+6«-8a''62'+T(J«' 
4^  =  V«  +  -TS-V 16tfi 


_  1/10  a<  -i-  ft*  -  16  fl*  _  i!^ 


312  TEXT-BOOK   OF   ALGEBllA. 


5. 

Solv^ 

e  V2  .7-  - 

-8)  = 

2 

,  2 

-8 

V2a;'^ 

■-16x- 

V2  x'  - 

2x^- 

2x''- 

Ux 

-  Wx 

-  4  X 

=  3() 

() 

-  12. T 

+  36 

which  is  a  quadratic  equation,  whose  solution  we  are  not 
now  in  position  to  obtain. 

328.  Rules  and  Directions  for  solving  Equations  involving 
Radicals. 

No  general  rule  can  be  given.  The  advantage  of  devia- 
tions from  the  normal  process  is  here  even  more  marked 
than  in  simple  equations.     (See  217.) 

1.  When  but  one  radical  appears  in  an  equation,  transpose 
all  the  other  quantities  to  the  opposite  side  of  the  equation, 
and  then  raise  both  sides  to  the  power  indicated  by  the  index 
of  the  radical. 

2.  When  two  or  more  radicals  occur,  it  is  usually  best  to 
place  one  alone  on  one  side  and  then  raise  both  members  to 
the  desired  power. 

3.  The  process  just  described  will  often  have  to  be  re- 
peated one  or  more  times  in  the  same  problem  before  all  the 
radicals  disappear. 

a.  When  fractions  occur,  it,  is  generally  best  to  clear  of  fractions 
before  squaring  or  cubing  or  raising  to  other  powers  as  the  case  may 
be.  Sometimes  an  advantage  is  gained  by  reducing  a  fraction  to  its 
lowest  terms,  or  by  rationalizing  a  denominator,  or  by  transposing 
and  uniting,  etc. 

h.  Care  should  be  taken  to  combine  all  similar  terms  before  rais- 
ing to  a  power. 

c.  By  example  3  the  student  is  admonished  that  the  value  of  the 
unknown  quantity  should  be  verified  in  tl\e  original  equation. 


iM>\vi:i:s,   uoo'is.  and  kaiucals.  'US 

329.    Exercise  in  the  Solution  of  Equations  containing  Rad- 
icals. 

1.    \  .,•  -  r>  =  :;. 


2.  r  -f  L'  —  V  H)  4-2-^  =  0 

3.  J-  =  9  —  VxM^i). 

4.  8-}-  V3^~-f0  =  14. 


6.    V2a;-f-ll  =  Vs. 


6.  Vr  -f~2r)  =  1  -f-  V.r. 

7.  v.r  -fy^  — .1-  —  .\  =  0. 

8.  V;^-"4  4-  '^  =  Va--f  11. 


9.    V4  ic  -|-  5  —  V^  =  Var  +  3. 


10.  s/ax  -\-2ab  —  n  =  l>. 

11.  V3  -^j^  —  A/3+li^=  0. 

12.  3  +  ^ar»-9ar'^  =  r. 


13.  V4  a  +  X  =  2  VA  +  >•  —  V.r. 

14.  V4  —  ViP*^*^'^  =  .r  -  2. 

15.  V9ar  _4-|-().=  S. 


16.    V4  .r«  -  7  ^  H-  1  =2  ./•  -  1 1 


17.     \  ./•  -h  v'4  -f  .1-  =     ^   • 


45 


18.  ..  +  V9  +  .«^^. 

19.  V.'-  4-  •'  +  v.'-  +  «  -  V4  rr'-fTM  =  0. 


20.     \  ./•  +    \  "         \  'IX  4-  ar*  =  V''. 


21.  = 


22.    \  y^/r  -f  ;5  ^    \  \  /  —  :;  =  V2\  /• 


314  TEXT-BOOK    OF   ALGEBKA. 


23.  =5V^-8  +  -^r- 


1    ,   1       ,/l  /  4         -9 


25.    -  +  - =  T- 


3/7^ 


26.    V64  +  ic^  -  8  a:  = 


4  +  a; 
a/4'+^' 


3  0-  —  1      '      ,  V3  X  —  1 

V.r)  X  -\rl  ^ 


FOURTH    GENERAL    SUBJECT. —QUAD- 
RATIC   EQUATIONS. 


CHAPTER   XXI. 

QUADRATIC    EQUATIONS    CONTAINING  ONE    UNKNOWN 
QUANTITY. 

330.  Quadratic  Equations  are  equations  of  the  nerrrmf  ilc- 
(free.     There  are  two  kinds.  Complete  and  Incomplete. 

Complete  quadratic  e(iu{itions  (also  called  (idfected  (piad- 
i-atic  equations)  contain  both  the  second  and  the  first  powers 
of  the  unknown.  Incomi)lete  quadratic  equations  (also  called 
inwe  quadratic  e(iuations)  contain  only  the  second  jx>wer. 

Thus,  itx'  -\-  bx  =  c  is  a  complete  quadratic  equation, 
while  inx^  =  n  is  an  incomplete  quadratic  equation. 

«.  These  are  the  only  eases  to  l)e  considered.  For,  the  first  equa- 
tion contains  the  s<iiiare  and  the  first  power  of  the  unknown  and  a 
term  which  does  not  contain  it.  There  can  be  no  other.  Tliis  equa- 
tion reduces  to  the  second  wIkmi  /»  (tlie  coefficient  of  x)  equals  zero, 
giving  ax'-  =  r,  a  pure  quatlratic;  if  a  —  0,  tlie  equation  reduces  to 
2&x=  c  whicli  is  no  longer  a  quadratic;  while  if  c  =  0,  it  reduces 
to  ax^  +  2  6x  =  0,  and  x  divides  out  leaving  ax  -H  2  6  =  0,  which  is 
also  a  simple  equation. 


315 


316  TEXT-BOOK    OF   ALGEBRA. 

SECTION   I. 
Solution  of  Incomplete  Quadratic  Equations. 

331.    Method  of  Solution  of  Incomplete  Quadratic  Equations. 

The  type  form  of  these  equations  is  mxr  =  n,  i.e.,  a  term 
containing  x''^  and  a  term  independent  of  x. 
The  sohition  is  very  simple. 

mx-  =  n  (Hyp.) 

n 
x'  =  -  (Ax.  4) 


.T  =  i  v/-  (A^-  6) 


a.    It  is  not  necessary  to  prefix  ±  to  both  .members  of  the  equa- 
tion, for  doing  so  merely  duplirafes  the  roots. 


Thus,  while  ±x  =  ±  i  /-  gives  rise  to  four  equations,  viz., 
Y'  m 

(1)  +  .r  -  +  4 /--'L    (2)    +x  =  -  t /a  (8)  -x=  +  J± 
\    in  \    ni  \    m 

■      (4)  -.  =  -./!. 

\    in 
two  of  these  equations  are  identical  with  two  others.    Eqs.  (1)  and  (4) 
are  the  same,  as  may  be  seen  by  multiplying  (4)  through  by  —  1. 
So  also  (2)  and  (3)  are  identical, 

1.  Given  i  (x''  -  10)  -f-  ^\  (6  .x'  -  100)  =  3  x''  —  6o  to 
lind  X. 

10  (.r2  -  10)  +  3  (6  X-  —  100)  =  90  x'  —  1950  (Ax.  3) 
10  .r2  _  100  +  18  x'  —  300  =  90  x'-  —  1950 

_  02  ./ -  =  —  l.V)()  (Ax.  a,  Sf)) 

x'  =  25  (Ax.  4) 

./•  =  i  V5.    A?is.  (Ax.  0) 

Yeisifkations.  Substituting  either  +  5  or  —  5  in  the  given 
equation,  i  (2.")  —  10)  +  J,  ( l-lO  —  100)  "-  7:,  —  (55  since  whether  we 
take  +  .")  or  — .").  when  squared,  it  gives  tlie  same  number. 


QUADKATIC    K<,)rATlONS.  ^U 


ax        X        d 


ax^  -f-  f^-"^^  =  ''^^^  —  '^^c? 
(a  4-  </)  A-"^  =  rr^/  (c  —  i) 

x-^=  «^  (c  —  b) 
a  -\-d 


=±v/'^ 


ad  (c  —  6) 


3.    Vx~+a  =  ^x  -\-  V^»2  4.  ar2 
J-  4-  ^.  =  ar  4-  V^''  +  g^ 
</  =  V**  4-  x^ 

_  ar'^  =  b-^  _  «2 

x"^  =  a'  —  b^ 

a:  =  -J-  Va^— ft*. 

332.    Exercise  in  the  Solution  of  Incomplete  Quadratic  Equa- 
tions. 

1.  3a:«— 2  =  2j'=4-2. 

2.  a;2_;^(;_  '^'4-12. 

9 
'4.    (9  4-a-)(9-j-)  =  19. 
6.    (0-4.  1)2  =  2  a- 4- 17. 

6.  4x  — ir)0.r-i  =.r— 3aj"^ 

7.  aV-  — //4-  o-=0. 

1  4-a-^l  —  j: 
-     a^  —  x'^ a 

'  x^—h'  ~~T' 


318  TEXT-BOOK    OF    ALGEBRA. 

4  {x''  -  5)       1  _  3(25-^'-^) +  10 


8               12       -"■'■               4 

11. 

1  4-  a^       a;  +  25 

l~x      ic  — 25* 

12. 

'"       b  +  ^.:ij 

1? 

^  -j- «       it-  —  a       ,^          4  a^ 

x  —  a^x-\-a       ^  ~  2a-\-l' 

14. 

(4  +  ^)  (5  -  7^)  +  23  =  (9  -h  2  a-)  (2  - 
4 

-3ic). 

1.^1 

','  _|_  AZ-y'^         17  — 

■'+^-'^       ''        V^^-17- 

18 

x  —  m       n  —  X 

X  -\-  m       n  -\-  x' 

17. 

ic-^  +  4  :  x^  -  11  : :  100  :  40. 

18. 

^x'-lx'-^:  I  x^  -  \  x"  +  3  : :  9  :  3. 
SECTION   II. 

SOI.ITTION    OF    (O.MI'LKTK    (^[   ADHATK:    EqT  ATIONS. 

333.  Complete  Quadratic  Equations  are  solved  by  a  method 
called  completing  the  square,  which  makes  their  solution 
different  from  anything  we  have  yet  had. 

1.  The  ec^uation  {(xx  —  by  =  c-  is  a  complete  quadra,tic, 
as  may  be  seen  by  squaring  the  left  member.  Moreover,  it 
can  be  solved  somewhat  like  an  incomplete  quadratic,  viz., 
by  extracting  the  square  root  of  both  members. 

{ax  -by'  =  e^  ■  (Hyp.) 

ax  —  b  =  ^c  (Ax.  6) 

ax  =  b  4-c  (Ax.  1) 

^  =  -^  (Ax.  4) 


QUADRATIC   EQUATIONS.  819 

2.  This  suggests  the  question,  can  all  complete  quadratics 
he  ])nt  in  the  same  form  '.'  A  little  consideration  will  show 
that  they  can.  Thus,  evnv  ((iinplete  (quadratic  eciuation  can 
be  reduced  to  the  type  Inini. 

This  means  that,  it  necessary,  the  e(iuation  is  cleared  of 
fractions,  the  terms  transposed,  collected,  and  arranged  so 
that  the  term  containing  x^  is  first,  that  containing  x  is  next, 
and  the  known  terras  form  the  right  member. 

Thus,  in  the  equation 

-7.    1        18a:\ 

=  llM2-h72.r  (111) 


3 

Cr  — 

<>)  ix  + 

^)  = 

IS  X- 

-72x- 

-2i<; 

ISx' 

-  144  ./• 

=  1 1 

<t 

=  IS. 

2  A  =  - 

-  144, 

and  so  with  any  other  ecjuation. 

3.  To  show  that  the  type  e(|uali(ni  nj-  -\-  2  hx  =  c  can  be 
reduced  to  essentially  the  same  form  as  {ax  —  h)'^  =  c'-^,  and 
can  then  be  solved,  we  j)roceed  as  follows  :  — 

ax''  -f  2  hx  =  r 

a^x'  -h  2  ahx  =  an.  (A  x.  .'{ ) 

Now  the  left  meml)er  consists  of  the  first  two*  terms  of 
the  trinomial  s(piare  of  f.r  -}-  A.  and  all  it  needs  is  tlie  third 
term,  b'\  to  make  it  a  jn  rt<ci  s.juare  as  desired.  But  there 
is  no  reason  why  we  mav  not  add  b'^to  the  left  member  pro- 
vided we  also  add  it  to  the  right  member. 

«V2  ■i-2abx-\-h^^b'-\-  ac (Ax.  \) 

ax-{-b  =  4z  V^"^  4-  ac     _  (Ax.  (5) 

X  = (Axs.  2  and  4) 

a  ^ 

We  concdude  that  every  complete  (piadratic  can  be  re- 
duced to  the   form  (ux  :^b)' =  b- ^^  ac,  and   can  then   be 


320  TEXT-BOOK   OF    ALGEBRA. 

solved  (after  extracting  the  square  root  of  both  sides,)  as 
two  simple  equations. 

Two  points  in  the  reduction  from  the  type  form  deserve 
especial  attention,  viz.,  making  the  coefficient  of  x"^  a  per- 
fect square,  and  completing  the  square. 

334.  The  Coefficient  of  x-  must  be  a  perfect  square  cmd 
2)ositive.  If  the  coefficient  of  x'^  is  not  already  a  perfect 
square,  as  1,  a^,  64,  9  h'-,  or  the  like,  the  equation  must  be 
multiplied  or  divided  through  by  such  a  quantity  that  the 
coefficient  of  x'  is  made  a  perfect  square  and  positive.  The 
extremes  in  a  trinomial  square,  as  is  evident,  must  be 
positive. 

335.  Completing  the  Square.  The  terms  added  must  be  so 
chosen  as  to  unite  with  the  other  two  to  form  a  trinomial 
square.  (See  135,  1.)  Let  us  investigate  how  this  term 
may  with  certainty  be  found. 

If  in  the  expression 

A^  -f  2  AB  +  B2 

Ave  have  A^  +  2  AB  given  to  find  B^,  the  problem  is 
easily  solved.  For,  dividing  2  AB,  the  second  term  by  2  A, 
twice  the  square  root  of  the  first  term,  the  quotient  is  B, 
which  must  be  squared  for  the  third  term.  To  illustrate 
this  let  us  take  the  5th  example  in  135,  1. 
Given  9  cc^  +  30  cc  to  find  25. 

2  V9^  =  6  x- ;  30  ic  -J-  6  ^  =  5  ;  o'^  =  2b. 

Given  4«^a-^  —  12  ahxy  to  find  9  h'^i/. 

2  V4  «-^:^'^  =  4  ax  ;  12  ahxy  -^ -iax  =3by;  {^  byf  =  9  by. 

As  an  exercise  in  this  process,  the  student  should  solve 
all  the  problems  of  135,  1  in  the  same  way. 

It  is  easily  seen  that  whe7i  the  coefficient  of  x^  is  1,  the 
process  gives  the  square  of  half  the  coefficient  of  x,  as  tlie 
third  term.     See  Ex's  1-4,  11-13,  16,  19  in  135,  1. 


QUADRATIC   EQUATIONS.  321 

336.    Examples    of    the    Solution    of    Complete    Quadratic 

Equations. 

('> 

1.  Given  1^ >  =  4 + 

x 

2  ./••- =  4  u- +  (>  (Ax.  ,S) 

j-^  —  'Jx=  :i  (334) 

2  Vx^=  '^  ./• ;    2  .r  -T-  2  u-  =  1 ;      1-  =  1  (335) 

x--i_2.r  +  1  =;{  + 1    =4 

j:  —  1  =     i  2  (Ax.  u) 

i.e.  J-  —  1  =     +2,  or  j—l  =  —2 
.  .  ./  =  ;^,  or  j:*  =  —  1.     .///.s'. 

VkKIKH  ATlo.N. 

1st.  X  =  »S  ill  the  given  ec^uatiou,  2  X  .'i  =  4  +  ♦;  i.e.  6  =  6. 
2(1.    ./•  =  —  1  in  the  given  equation,  2X—  1  =  4  +  -^ , 
i.e.,  _  2  _.  —  2. 

2.  Solve  i)0  x'^  —  15  x  =  27 

100^2  _  3() ^  ^  54  (334, 

2  VlOO  r-«  =  20  X ;  30  X  -i-  20  -p  =  ^  ;  (•;!)-  =  i( 
KM)  X-  _  30  J-  +  I  =r  54  +  24  =  mi  =  5(>.25 
10  j:  —  I  =  -1-  7i  i.e.  +  7i  an<l  -  7},. 
.-.     10x  =  ^  +  7i,  or  ij  -  7.1 

^  =  Aj  «!•  -  H>  ==  -  5-     '^"•^■• 

3.  Solve  (x  4-  1)  (-'^  +  3)  =  4  j:'^  _  22 

2x2 -f.  5a:  4.  3  =  4^^-22 
—  2«2-f-5a;  = -25 
a;2  _  ^  jc  =  _j_  :;^^. 
f-2^1;  (t)-^  =  fg 

«' -  B^ -h  f  ^  =  ¥  +  e  =  Vc*'^ 
a:  -    ^  =  i  Y 

X  =  5,  or  -  2.V. 

4.  Solve     3x^-\-    10  a:  =    57 

36^2  -f- 120  a-  =  6S4  (334) 

2  V36  3^2=    12  X ;  120  a-  -^  12  a-  =  10 ;  10'^  =  100 
36a;«  +  120a;  +  l<W)  =  784 
6a;+    10  =  ^28 
X  =  3  or  —  'Y- 


322  TEXT-BOOK  OF  ALGEBKA. 


Solve      nx^  -\-px  =  q 
4  n'-x'  +  4  7i2)x  =  4  nq  (334) 


4  n^x'^  -{-  4  7i2)x  =  4  nq 
2  V4  Ti^a?^  =  4  7ix  ;  4  «^9j;  -i-  4:7ix  =  p', 
4  w^ic^  +  4  7^79^  +  7>'^  =  4:nq  -\-  p^ 
2nx  -\-  p  =  ^  V4  ^j.g'  +  pi^ 
—  p^  V4  Tiy  +  p^ 

X  =  7. 


(py-p' 


2n 

Observatiox.  —  This  example  shows  that  if  an  equation 
be  multi2:)lied  through  by  4  times  the  coefficient  of  x^,  the 
square  of  the  old  coefficient  of  x,  viz.,  p,  is  the  quantity  to 
be  added  to  complete  the  square.     See  also  Ex.  4. 

337.    Rules  for   Solving  Complete  Quadratic   Equations.  — 

One  rule  only  is  really  needed,  the  first  and  second  rules  as 
given  below  being  special  cases  of  the  general  process  ex- 
plained in  the  third  rule. 

1.  First  rule. 

(1.  Reduce  the  given  equation  to  the  type  form  ax"^  -\- 
bx  =  c,  and  divide  through  by  the  coefficient  of  x^,  if  it  is 
not  already  1. 

(2.  Add  the  square  of  one  half  the  coefficient  of  x  to 
both  sides  of  the  equation  and  extract  the  square  root  of  the 
two  members. 

(3.  Solve  the  two  resulting  simple  equations  for  the 
two  values  of  x.    (See  Examples  1  and  3  of  the  last  article.) 

2.  Second  rule. 

(1.  Reduce  the  equation  to  the  type  form  (removing 
any  monomial  factors  which  may  exist  in  it,  Ax.  4)  ;  then 
multiply  through  by  4  times  the  coefficient  of  x'^,  merely  in- 
dicating the  multiplications  in  the  left  member. 

(2.  Add  to  both  sides  of  the  equation  the  square  of  the 
old  coefficient  of  x,  and  complete  the  solution  as  in  rule  1. 
(See  Exs.  4  and  5.) 


QUADRATIC    EQUATIONS.  '>-•> 

3.  Tliinl  rul(*. 

li.  Reduce  the  e(|uaLi«»ii  to  llie  t  \  pt-  Inini  tiiid  umltiplv 
or  divide  tlirough  by  such  a  (juautity  as  will  make  the  coef- 
ficient of  x'^  a  iMii'fect  square,  and  at  the  same  time  shorten 
to  the  greatest  extent  the  subsequent  reductions. 

(2.  Add  to  both  sides  of  the  equation  such  a  quantity 
as  will  make  the  left  member  a  j)erfect  trinomial  scjuare. 
To  do  this  ilivide  the  second  term  by  twice  the  square  root 
of  the  first  term,  and  square  the  quotient  for  the  third  term. 
Complete  the  solution  as  in  the  first  rule. 

rt.  The  first  and  second  rules  are  easier  for  beginners.  The  rule 
just  given  is  really  the  best  because  it  is  most  flexible.  However, 
considerable  experience  is  needed  to  be  able  to  use  it  to  the  best  ad- 
vantage, .Suggestions  will  be  made  from  time  to  time  to  show  the 
student  its  sui)eriority  over  tlie  others,  and  tlierefore  the  desirability 
of  mastering  it  as  well  as  the  others.  The  second,  which  ensures  the 
avoidance  of  fractions,  is  sometimes  called  the  Hindoo  Rule. 

4.  Fourth  rule. 

(1.    Reduce  the  equation  to  the  type  form. 

(2.  Substitute  the  values  of  n,  p,  and  q  from  the  given 
j)roblem  in  the  answer  of  example  5  of  the  last  article. 
(See  255.) 

Thus,  given    y  -  ^  +  '  ^  =  8 

12  .r'^  —  8  .r  =  15  .  (to  type  form) 

nx'^  +  jw  =  y  (Eq.  of  Ex.  5) 

So  here,  «  =  12 ;  yy  =  —  8 ;  and  «/  =  15. 

Substituting  these  values  in  the  value  of  x  found  in  l^x.  5. 

^  _  -  (-  8)  -i-  V64  +  4  X  12  X  15 
2X  12 


.    ^j.  -  8  (-  8)  -  V64  +  4  X  12  X  15 
2X12 
«  ^  +  ^'8  ^  .3     ^^^.  8  —  28  ^  _  5 
"*      24  •_'*   "'    '    24  (5 


Ans. 


324  TEXT-BOOK  OF  ALGEBRA. 

338.  Exercise  in  the  Solution  of  Complete  Quadratic  Equa- 
tions. 

1.  x'^  —16  =  45  —  4  X.        7.    X-  —  341  =  20  x. 

2.  a;2  +  22  ic  =  75.  8.    23  ic  =  120  +  x\ 

3.  a;2  =  X  +  72.  9,    x^  —  6x  =  6x-\-  28. 

4.  3  ?/2  +  48  =  30  I/.  10.    4  o;-^  +  4  x-  =  -  1. 

5.  5  ./•-  4-  20  X  =  25.  11.    ^2  —  *^  ;;>•  =  32. 

6.  x^  —  () ./'  =  0.  12.     '"  4-  350  —  12  /■  =  0. 

10  ^ 

,_     .T+22       4  _  9^-6 
j_^, —  —  —  , 

3  X  2 

14.  (./•  —  1)  (X  —  2)  =  1. 

,,     19  11 

15.  __  a-  =  —  —  ic^ 

5  5 

16.  Sa;'-^  — 20x  =  21. 
Suggestion.  —  Multiply  through  by  2. 

17.  ^'  +  i^x  =  20 .r  +  25J  -  ^^x\ 

Suggestion.  —  After  clearing  divide  through  by  6. 

18.  252 x"-  +  360  x=  —  125. 

Suggestion.  —  252  =  7  x  36.  Therefore  if  an  extra  factor,  7,  is 
introduced,  the  product  is  a  perfect  square.  Multiplying  the  equation 
through  by  7,  we  get 

7  X  252  x2  4-  7  X  360  x  =  -  875 


2  ^7'-^  X  36  x2  =  2  X  42  x;  7  X  360  X  ^  2  X  42  X  =  30;    (30)^  =  900. 

(42  x)2  +  (    )  +  (30)'-^  =  900  -  875  =  25 
42a;  +  30=-t5 

x  =  -  If,  or,  -  f 

(        ^l^  —  15#^  =  —  815  =  —  125 

Vek.k.cation.  -  \       j^.  _  3^,^  ^  _  j|._ 


i^UAUUATTC    EQFATTOXS.  •>-•> 

Kem AKK.  —  To  see  the  advantage  gained,  the  student  should  solve 
this  exercise  by  the  first  and  second  rules. 

19.  72  y- +  40S  r  =  1222. 
Si'OOESTiox.  —  Divide  through  by  2. 

20.  84  a-'^ -f  45  =  124  J-. 

Suggestion.  —  84  =  7  x  :J  x  4.  Hencemultiply  through  by  7x3. 

21.  96  x*  =  4  jr  -f  15. 

Suggestion.  —  tMi  =  (i  X  US.     Hence  multiply  through  l»y  «*». 

22.  622  J-  =  15  x'  -f-  6.3<S4. 
Suggestion.  —  Multiply  through  by  lo. 


26. 

-  =  2  ox  —  cx^ 

27. 

l<;.r-i_4  =  12  a- ^. 

23.  l{:t  Jt^)  (^  -  -0  =  ii  f  ^A  +    ^^-  j 

24.  :ix-\-  4  =  39j;-^ 

X       n       2 

25.  --!--=-. 
ff    '   X       a 

1         4 

28..  +  ^  =  ^. 

Suggestion.  —  Multiply  through  by  r.  and  not  by  x  \/i) 

29.  986  a-  — 14508(1  =  x\ 

30.  7  x'^  _  7  .r  =  1 . 

31.  (ixr-hi  =  (v^o'-i!i-^-a->-)' 

SiGGESTioN.  —  Transpose  and  unite,  without  clearing  of  frac- 


)ns. 

32. 

J '  —  {ft  -\-  ^0  ^  -\-  "^  =  ^*- 

33. 

X 

34. 

(m  —  7i)  x^  —  nx  =  VI. 

35. 

1               1           _  3  -h  rr* 
a  —  X       a  -f-  '  ~~   a*  —  ^* 

36. 

1     ..  Ill 

„  —h  4-  .r         n         1,  ^  .,' 

326  TEXT-BOOK   OF   ALGEBliA, 

37.  Vx  -1  =x—l. 

38.  X  —  Vx  =  20. 


39.  V 2  X  +  2  +  VT  +  6  ;:c  =  V7  a?  +  72. 

40.  7  (a:  +  7)  +     ^      J       ^  =  0. 

41.  9  a'b^x^  -  6  a%^x  =  b'\ 


42.  V  :c  +  <^  —  ^x  —  a  =  V2  X. 

43.  (^a?'-^  —  2  ex  y/d  =  ax^  —  cc^. 


SECTION  III. 

PROBLEMS. 

339.  Problems  Involving  the  Solution  of  Quadratic  Equa- 
tions, Complete  and  Incomplete. 

a.  Algebra  is  a  formal  science  made  to  cover  all  cases,  and  with- 
out any  reference  to  particular  problems.  Some  problems  by  their 
nature  admit  of  negative  numbers:  in  such  a  negative  answer  has  its 
proper  significance;  while  in  others  negative  values  for  the  answer 
are  inadmissible.  Moreover,  in  algebra,  imaginary  values  for  the 
unknown  denote  that  the  problem  is  hupossible. 

1.  Find  two  numbers  one  of  which  is  5  times  as  great  as 
the  other,  and  the  difference  of  whose  squares  is  90. 

Suggestion.  —  Let  x  =  the  less,  then  5x  =  the  greater. 
Then  the  equation  is  (5  xy  —  x^  =  96. 

2.  The  square  of  a  certain  numljer  diminished  by  17  is 
equal  to  130  diminished  by  twice  the  square  of  the  number. 
Required  the  number. 

3.  A  person  bought  a  quantity  of  cloth  for  $120;  and  if 
he  had  bought  6  yards  more  for  the  same  sum  the  price  per 
yard  w^ould  have  been  $1  less.     What  was  the  price  ? 


(ji  AhK.vric  i:(jUATioxs.  327 

Let  X  =  the  price  per  yard,  then  the  equation  is 

l^     -£-\.     Therefore  X  =  5  or  -  4.  (219,2) 

X 

But  —  4  can  have  no  meaning  in  this  problem.  (253).  We  can, 
however,  so  modify  the  statement  of  tlie  problem  as  to  be  able  to 
make  use  of  the  second  answer.  Thus,  —  The  exchange  account  of 
a  banker  amounted  in  a  certain  number  of  days  to  $120,  during 
which  time  exchange  remained  the  same.  Had  the  period  differed 
from  what  it  was  by  0  days,  and  in  his  favor  (i.e.  0  more  if  premi- 
ums and  0  less  if  discounts)  it  would  have  made  a  difference  of  $1 
per  day.  What  was  his  daily  premium  or  daily  discount  ? 
Let  X  =  the  daily  premium  or  discount. 

Then  referring  to  the  equation  above,  we  see  that  if  x  is  positive 
the  statement  is  satisfied;  also  if  x  is  negative,  ^  is  negative  and  6 
added  diminishes  the  number  of  days  numericaUij.  The  two  answers 
5  and  —  4  signify  a  daily  premium  of  $5  or  a  daily  discount  of  $4. 

4.  Find  a  number  such  that  if  you  subtract  it  from  10  and 
multiply  this  luimber  by  the  number  itself  the  product  shall 
l)e21. 

5.  Divide  the  number  346  into  two  such  parts  that  the 
sum  of  their  square  roots  shall  be  26. 

Let  X  =  the  square  root  of  one  part,  and  26  —  x,  that  of 
the  other. 

6.  A  merchant  bought  a  piece  of  cloth  for  $.324,  and  the 
number  of  dollars  he  paid  for  a  yard  was  to  the  inunher  of 
yards  as  4 : 9.     How  many  yards  did  he  buy  ? 

7.  If  a  certain  number  be  added  to  94  and  then  the  same 
number  be  taken  from  94,  the  product  of  these  derived  num- 
bers is  8512.     What  are  the  numbers  ? 

8.  A  man  traveled  60  miles  and  if  he  had  traveled  1  mile 
an  hour  more  he  would  have  required  3  hours  less  to  per- 
form the  journey.     At  what  rate  did  he  travel  ? 

9.  Find  three  consecutive  numbers  whose  sum  is  equal  to 
the  product  of  the  first  two. 


328  TEXT-BOOK   OF    ALGEBRA. 

10.  A  rectangular  field,  whose  length  is  3367  and  whose 
breadth  is  37  yards  has  by  it  another  field  of  an  equal 
number  of  acres  whose  length  is  to  its  breadth  as  13  :  7. 
What  are  the  dimensions  of  the  latter  ? 

11.  An  individual  bought  a  certain  number  of  kilograms 
of  salt,  4  times  as  many  of  sugar,  and  8  times  as  many  of 
coffee,  and  paid  for  each  of  them  40  times  as  many  cents  as 
there  were  kilos  of  that  material.  How  many  kilos  of 
coffee  did  he  buy  if  he  paid  altogether  $32.40  ? 

12.  If  a  certain  number  is  diminished  by  3  and  also  in- 
creased by  3,  then  is  the  sum  of  the  quotients  which  we  get 
by  dividing  the  greater  by  the  less,  and  the  less  by  the 
greater  equal  to  3^y.     What  is  the  number  ? 

13.  A  capital  stands  at  4^  interest.  If  the  number  of 
dollars  of  capital  be  multiplied  by  the  number  of  dollars  in 
5  months'  interest,  the  product  is  10o3375.  How  many 
dollars  are  there  in  the  capital  ? 

14.  The  hypothenuse  of  a  certain  right-angled  triangle  is 
2  ft.  greater  than  the  base,  and  9  ft.  greater  than  the  per- 
pendicular.    Find  the  sides  of  the  triangle. 

15.  A  girl  bought  a  number  of  oranges  for  40  cts.  Had 
she  bought  at  another  place  she  would  have  received  3 
more  oranges  for  the  same  money,  each  orange  costing  §  of 
a  ct.  less.     How  many  did  she  buy  ? 

16.  Two  peasant  women  together  brought  260  eggs  to 
market,  and  both  lost  the  same  amount.  "  Had  I  sold  your 
eggs,"  said  the  first  to  the  second,  ''  and  had  they  brought 
my  price,  I  should  have  lost  on  them  7.20  marks."  ''  That 
may  be,"  responded  the  other,  "  but  if  I  had  sold  your  eggs 
at  the  price  mine  brought,  I  should  have  lost  9.<S0  marks." 
How  many  eggs  did  each  bring  to  market  ? 

Suggestion.  —  The  equation  is  x.  -~^ =  ' '      (260  —  a*), 

2b0  —  x         X     ^ 

which,  after  clearing,  may  have  the  square  root  of  each  memher  ex- 
tracted ((s  if  sfands. 


QUADRATIC    K(a'ATIONS.  829 

17.  The  i)eriineter  of  a  rectangular  field  is  500  yas.,  and 
its  area  14400  sq.  yds.     Find  tin-  length  of  the  sides. 

18.  What  are  eggs  a  dozen  wlieu  two  more  in  a  shilling's 
worth  lowers  the  price  a  penny  per  dozen  ? 

19.  Find  two  numbers  whose  difference  is  rf,  and  whose 
product  is  }/. 

20.  Divide  a  straight  line  a  inches  in  length  into  two 
parts  such  that  the  longer  part  may  be  a  mean  proportional 
between  the  whole  line  and  the  shorter  part. 

This  is  called  in  geometry  dividing  the  line  in  "  golden 
section,"  or  into  mean  and  extreme  ratio.  (See  378,  1) 

21.  What  number  is  that  whose  square  increased  by  5  is 
givater  by  23  than  7^  timos  tlio  number  ? 

22.  The  product  of  two  nunil)ers  is 79  and  their  quotient 
is  q  ;  re(iuired  the  numbers. 

23.  The  sum  of  the  scpiares  of  two  numbers  is  c,  and  their 
difference  is  d.     Required  the  numbers. 

Wliat  hap]>ens  if  2c  is  less  than  d^  ?  Thus,  e.g.,  take 
d  =  <)  and  2r  =  SO.     (See  Art.  265.) 

24.  Two  iK)ints  start  out  together  from  the  vertex  of  a 
right  angle  along  its  respective  sides,  the  one  moving  m  ft. 
l)er  second,  the  other  n  ft.  jmi  s.  ( ond.  How  long  will  they 
recpiire  to  be  r  ft.  apart  ? 

26.  It  is  required  to  tind  three  numbers  such  that  the 
product  of  the  first  and  second  equals  a,  the  product  of 
the  first  and  third  ecjuals  h,  and  the  sum  of  the  .squares 
of  the  .second  and  third  equals  r. 

26.  A  set  out  from  C  towards  I)  and  travelled  7  miles 
l>er  day.  After  he  had  gone  32  miles,  B  set  out  from  I) 
towards  ( '  and  went  every  day  ^\i  of  the  whole  journey,  and 
after  he  had  tmveled  as  many  days  as  he  went  miles  in  a 
day,  he  met  A.     Required  the  distance  from  C  to  D. 


330  TEXT-BOOK   OF    ALGKBUA. 

27.  A  reservoir  can  be  filled  by  two  pipes,  by  one  2  hours 
sooner  than  by  the  other.  If  both  pipes  are  open  at  the 
same  time  the  reservoir  is  filled  in  1|  hours.  In  how  many 
hours  can  it  be  tilled  by  the  smaller  pipe  alone  ? 

28.  A  farmer  sowed  one  year  a  hektoliter  of  wheat ;  the 
next  year  he  sowed  what  he  harvested  the  first  year,  less  b 
hektoliters,  and  reaped  c  fold  of  what  he  sowed  and  d  hek- 
toliters  beside.  Assuming  a  like  fruitfulness  both  years 
how  much  did  he  reap  the  first  year  ? 

29.  It  is  required  to  divide  each  of  the  numbers  21  and 
30  into  two  parts,  so  that  the  first  part  of  21  may  be  3 
times  as  great  as  the  first  part  of  30 ;  and  that  the  sum  of 
the  squares  of  the  remaining  parts  may  be  585. 

30.  A  looking  glass,  in  size  a  X  b  inches,  has  a  frame  of 
uniform  width  and  of  the  same  area  as  the  glass.  What  is 
the  width  of  the  frame  ?  Suppose  a  =  12,  and  b  =  18,  i.e., 
a  glass  12  X  18,  and  substitute  in  the  answer  as  by  the 
fourth  rule. 

31.  A  square  courtyard  has  a  rectangular  walk  around  it 
(on  the  inside).  The  side  of  the  court  is  2  yards  less  than 
six  times  the  breadth  of  the  walk ;  and  the  number  of 
square  yards  in  the  walk  exceeds  the  number  of  yards  in 
the  perimeter  of  the  court  by  92.  Find  the  area  of  the 
court. 

32.  The  driving  wheels  of  a  locomotive  are  two  feet 
greater  in  diameter  than  the  running  wheels  ;  the  running 
wheels  make  140  turns  more  than  the  driving  wheels  in  a 
mile.  What  are  the  diameters  ?  (The  circumference  of 
a  Avheel  =  3f  times  the  diameter,  nearly.) 

33.  Generalize  the  preceding  problem. 

34.  A  grazier  bought  a  certain  number  of  oxen  for  f  240, 
and  after  losing  3  sold  the  remainder  at  $S  per  head  more 
than  they  cost  him,  thus  gaining  $59.  What  number  did 
he  buy  ? 


(,>r.\i>i;.\iic    i.<,»i    \  ri<  "NS.  331 

35.  Generalize  tlie  iUtli  iirobleiiL 

36.  A  wall  is  ])iiilt  l)y  two  masons  of  which  the  first 
ht'<,niis  to  work  1.^  days  later  than  the  other.  It  takes  o^ 
(lavs  from  the  time  the  first  began.  How  long  would  it 
take  to  finish  the  wall  should  the  first  work  li  days  less 
than  the  second  ? 

SoMTioN.  —  Let  X  —  the  number  of  days  the  first  requires  in  which 
to  build  the  wall,  and  y  =  the  number  of  days  the  second  requires. 

Ihen   — u        =1    whence    -  = 

y^  X        ^    ^»ence    ^  ^^^ 

There  is  need  for  still  another  unknown  which  we  may  denote  by 
z.  Let  z  =  the  number  of  days  n'(|iiin(l  wluii  the  first  works  3  days 
less  than  the  second.     Then, 

J-  \      14x      / 

14  z- 42  4- 2x2-11  2  =  14  a-. 

Thus  it  appears  that  we  have  to  find  the  values  of  x  y,  z,  k 
8/«f//e"quadratic  equation  in  two  unknowns,  which  is  indeterminate. 
The  method  of  solution  will  be  like  that  of  247.     Solving  for  x, 

32-42  ,       z-28 


14-22  22-14 

To  make  x  positive  the  signs  of  the  two  terms  of  the  fraction 
must  be  unlike.  For  this  we  can  assume  z  =  8,  and  10,  both  of 
which  give  positive  integral  values  for  x,  but  only  2  =  8  gives  y  pos- 
itive and  integral.  Thus  we  have  2  =  8,  x  =  9,  y  =  18.  Of  course 
in  the  present  problem  there  is  nothing  to  interfere  with  z  having 
fractional  values  within  the  limits  determined  by  the  above  equation. 

37.  A  man  has  $130(),  which  he  divides  into  two  poi-tions 
and  loans  at  different  rates  of  interest.  If  the  first  portion 
had  been  loaned  at  the  second  rate,  it  would  have  produced 
$3G ;  and  if  the  second  portion  had  been  loaned  at  the  first 
rate  it  would  have  produced  ^49.  Required  the  rates  of 
interest. 


832  TEXT-BOOK  OF  ALGEBRA. 

38.  Required  the  rates  in  the  preceding  if  the  two  por- 
tions produce  equal  returns.  How  does  this  problem  differ 
from  the  preceding  ? 

39.  There  is  a  certain  fraction  to  each  of  whose  terms  if 
1  be  added  the  product  of  the  resulting  fraction  and  origi 
nal  fraction  is  |.     Required  the  fraction. 


QUADKATIC    EQUATIONS.  333 


CHAPTER   XXII. 

SIMULTANEOUS  QUADRATIC   KQUATIONS. 

340.  Simultaneous  Equations  in  Quadratics.  See  225.  Two 
cases  may  be  distinguished.  First,  when  two  or  more  of 
the  given  equations  are  of  the  second  degree  ;  and  second, 
when  there  is  but  one  quadratic  equation  given.  Perhaps 
the  latter  ought  to  be  called  simultaneous  quadratic  and 
simple  equations.     See  195. 

Thus,  -]  ?^2  T  *  '^  Z  ijj  1^  «i^^  example  of  the  first  case  ; 

while,     \  11  -^  y  +  *^  -^  +  1*;,^  //  =  }•>  iy  an  example  of  the 
second  case. 

SECTION  I. 

Simultaneous  Equations,  One  of  wnicu  is  of  the  Second 
Dkgkee,  and  tmk  Others  Simple  Equations. 

341.  Solution  of  Simultaneous  Equations  where  One  is  Quad- 
ratic.—  To  solve  such,  substitute  in  the  (quadratic  the  values 
obtained  from  the  other  equations,  so  as  to  get  a  single 
quadi-atic  equation  containing  one  unknown  quantity. 
Solve  this  and  substitute  the  roots  in  the  other  equations. 
Two  sets  of  answers  will  generally  result.  Examples  will 
make  this  plain. 

1.    Given(l)   a:^ -f  3a:y  =  54,  and  (2)  .r -f  4  y  =  23 
(20   a.  =  23-4y. 
(3)    (23  -  4.yy  4-  3  (23  -  4y)  y  =  r,4       .228, 


334  TEXT-BOOK    OF   ALGEBRA. 

.  (3,)    529  -  184  ^  +  16  >f  +  69  ?/  -  12  f  =  54 
(82)   4y2_ii5^_  _475  (334^ 


9     115 .   75 

-^  ^  ~  ~4~  ~  ^  X 

y  =  23f ,   or    j  _?/  =  5 
.\  a:'  =  —  72,  (  ic  =  3. 


(229) 


f  (1;  a:^  +  /  +  ^^  =  50 
2.    Given    -<[  (2)   2  .r  -f  3  //  +  .-  =  23 
1  (3)   ^4- 2./ +  3^  =  23 

Eliminating  x  and  y  (so  as  to  have  z  only  in  the  quad- 
ratic e(]uation)  from  equations  (2)  and  (3),  and  substituting 
their  values,  — 

(2,)   2ic  +  3v/  =  23-.~ 

(3i)   2  a-  +  4  7/  =  46  -  6  z 

(4)  7/=-23-5.^  (xVx.  2) 

(2o)    4  .X-  +  6  //  =  46  -  2  ,^ 
(32)    3:^-  +  6//:=  69  -  Si'z 

~~jc  =1  z-  23  (Ax.  2) 

(1)    (7  z  -  23)^  +  (23  _  5  ^)^  +  ^^  =  50  (228) 

49  z-  -  322  ;v  +  529  +  529  -  230  z^^z^  \z'  =  50 

75-2-  552.^;  =  _  loos 

25  -2  —  184  ;^  =  -  33() 

25^2-(       )  +  (18.4)v=  2.56 

^,,^'   _  1S.4  =  il.6 
[  ^  ==  4.  ,,1.  ^  3=  ;>_;>(;  ^ 

-;   (4)   //  =  3        //  =  ().20   ;       Ans.  (229) 

[  (5)   ./■  =  5        ./•  =    .52  j  (229) 

342.  Special  Forms  for  which  there  are  more  elegant  solu- 
tions tlrni  by  sidistitution.  —  If  the  two  given  (Mjuations  en- 


yiADKATIC    KgUATlONS.  335 

able  us  to  calculate  quickly  the  values  of  the  expressions  x- 
rlz- a*//  -|-  y^f  we  liave  innnediately,  by  extracting  their  square 
roots,  the  values  of  x  -j-  //  and  x  —  y^  from  which  the  values 
of  X  and  y  are  readily  found  by  adding  and  subtracting  these 
equations.     (255,  4.) 

1.    Given  (1)  x""  +  y^  =  74  and  (2)  x  -\-  y  =  12 

(Ax.  o) 


[(20  -  (30,  Ax.  2] 
(Ax.  6) 

(Ax.  1) 

(Ax.  2; 

=  r>  and  //  =  7. 

2.    {\)  X  ~  y  =.  12  (2)     ./•//  =  S5 

(1,)  X-  -  2  ./•//  +  //-  =  144  (Ax.  .')) 

(2,)  4xy  ^  340 

(3)  x^  -f  2  xy  4-  //«  =  4<S4  (Ax.  1 ; 

(3.)  .r  -h  y  =  i  -^-*  (Ax.  G) 

(l).r-y=        12 

.r  -^        17.  or  —  /) 

//  —         .">.      —  17 

(/.  A  verificHtion  will  show  in  this  as  in  all  the  ♦'xanii)l«'s  of  this 
and  the  prt'ct'ding  article  that  the  answers  are  ohtained  in  sets.  Tims 
+  17  and  +  .'>  go  together,  and  —  .j  and  —  17.  Hut  -I-  17  and  —  17 
as  values  of  j  and  y  would  not  satixfy  the  equations.  So  with  all 
<]nadratic  solutions. 


(-^) 

.c2-|-2.r//  +  /=144 

(1) 

x''               -fy^=    74 

(^) 

2xy          =    70 

(30 

4  xy          =  140 

(4) 

a:"-^-2ry +  //■-  =  4 

i^^) 

^-  -  //  =  i  -> 

{V 

^+.'/=    1-^ 

2x=  14  or  10 

.r=    7  or    5 

!>//  =  looi  14 

//  =    T)  or    7. 

A  US.    ./  =  7.  //  =  ."> ;  or.  ./• 

336 


TEXT-BOOK    OF   ALGEBRA. 


343.  Exercise  in  the  Solution  of  Simultaneous  Equations. 
—  Those  which  come  under  342  should  be  solved  by  the 
method  there  explained. 

1. 
2. 
3. 


x^y  =  l 

x' -{- 2  y' =  34.. 

6 

x-\-4.y  =  23 
x^  +  3  xy  =  54. 

7. 

5x  —  y  =  17 
xy  =  12. 

8 

\3x  -  y  =  11 
I  3  x'  -  /  =  47 

i  x-\-  !/  =  !'"> 
I  xy  =  3G. 


(3^-2y  =  2 

}9x'  +  iif  =  394. 

\  xtj  =  923 

(.•  +  2/  =  84. 

xy  =  -  2193 
x  +  !j=-^. 

<   J'     '    y 

10. 


\2x-\-y  =  l 
)4^^  +  /  =  25. 


11. 


'^^  +  f  +  ^'.y  =  20S 
(  ^  +  y  =  16. 

SuGGESTiox.  —  First  find  the  value  of  xy  and  then  solve 
as  in  the  last  article. 

x  —  y  =  3 

x^  -3xy-^tj^=  -  19. 

Suggestion.  —  Divide  the  first  equation  by  the  second  in  Ex.  13. 


12. 


13. 


14. 


15. 


ic2  _  ^2  _   16 

x  —  y  =  2. 


4.  (xhj'')  =  13  xy 
X  —  y  =  6. 


y    '   X         ^ 
x-\-y  =  6. 


16. 


17. 


18. 


[       ,1        2x-^y 


x-{-y 


3 
4  X  —  y 


[   * 


+  ^^^±1^0 


x  +  3 
[lx-^3y=l 

I  s  +  ^  =  ^ 


QUADRATIC   E(JUATIONS.  337 


19.  jc        ,  21.  '    '',       ..        ... 

y 

^^-      ^  {X  -  2)  -  (//  -  3)  =  b.      22-       (  ry^  +  r/y^  =  <y. 


23.      (1^:^::1'J:'^// 


SECTION  II. 


Simultaneous  Equations,  two  oh  mokk  of  wiik  ii  akk  ok 
TiiK  Sk<«)M)  Dk<;hkk. 

344.  Solution  of  Simultaneous  Quadratic  Equations.  —  It  is 
not  possible  to  give  a  geiuTiil  solution.  .Many  particular 
examples,  however,  can  be  solved  by  special  methods. 

1.   Given  (1)  y^  -  'J  jKi'i/  =  n^  and  (2)  y^  +  2  qxi/  =  h. 

(10    x^-2pxij+pY'^^'''^p->f  (333,3) 

(1.)     ^  =py  i  V«='  ^pY  (Ax.  (>) 

(20    y^  +  2  q{py  Jz  V^^  +  y>V)  //  =  f'  (228) 

i 2yy  V«^  +  pV  =  A  -  /  (1  +  2pq)  (328) 

4  qh/  («« -f  2>y)  =  ^.^  -  2  hi/  (1  4-  -'  />v)  +  //  (1  4-  2  pqY 

(Ax.  5) 

which  is  an  equation  of  the  fourth  degree  and  cannot  in 
general  be  solved. 

a.  The  general  equation  of  the  second  degree  in  two  unknowns, 
i.e.,  one  which  contains  every  possible  term,  is 

ax2  +  hxy  +  nj'^  +  <lx -^r  ey +f=0. 

If,  now,  two  such  equations  as  this  be  combined  the  process  of  solu- 
tion is  quite  like  the  example  just  reduced  (only  very  much  more 
complex),  and  therffon*  is  not  solvable  by  direct  algebraic  methods. 


338  TEXT-BOOK    OF   ALGKBKA. 

2.    G-iveii  (1)  .T-  +  //  =  a  and  (2)  x;/  =  h. 

(3)  x-'^  +  2  xy  +  //-  =  </  +  2  ^  (Ax.  1) 
(3i)  0^  +  //  =  i  V^r+ 2  />  (Ax.  6) 

(4)  ic'^  -  2  a-y  +  /  =  a-2  />  (Ax.  2) 
(40  0^  -  y/  =  ,i,  V^^  -  2  ^>  (Ax.  6) 

i  V«T^J7^  i  Va  -  2  /> 
.-.  X  =  -F= 


y  = ^ 


Remark.  —  The  sign  ±  (read  "minus  or  plus")  means  minus 
fit\st. 

3.  {I)  xy  +  Qx  +  1  ij  =  50.    (2)  3  if?/  4-  2  ^  +  5  v/  =  72 
(li)  3  .ry  +  18  ic  +  21  y  =  150 

(2)  3^y+  2.^+  5y=  72 

(3)  16  a;  +  10  2/=  78 

(3.)  ....  =  ^;-i-^ 

(l)^«^^..  +  ??iii^  +  7,  =  50     (228) 

39 y  -  8  y-^  +  234  -  48  //  -^my  =  400 
8  y2  __  47  y  =  _  166 

47    ■  166 

ir^=--8- 

^  ~  8^  +  (^16J  ~  256  ~  256 

47  ,  1   / 

^  =  16^16^-^1^^- 

(3)         ^  =  +f^=p^V^=-3103.      (229) 

4.  (1)  2ic2  3^^_|_y2^63_^  (2)  ;r2_^2ify  +  6//=  174. 
Put  y  =z  rx,  r  being  unknown. 

Tlien  substituting  the  value  of  y  in  each  equation. 


y^-^y  = 


Ql'ADliATIC    i:gl   A  ri<>N.>. 

(1)  2x^-^3rx'-\-     rV^=    63 

(2)  x^  -\-2  rx^  +  (•)  r-y-  =  174 
(1.)   (2-h.Sr  +  r^)u-^        =    G3 


..  +  ;-2 


330 


(97) 
(Ax.  4) 


,1.\,    ,^1  -f-L',--^  (•,/-)  .•^=174  (97) 

>              174  ,,      ,, 

03 1^4  ^ 

2T37T^  -  i+2r  +  6r2  ('*'J1,  Ax.  < ) 

63  +  126  r  4-  378  r*^  =  ;U8  +  522  r  +  174  r*  (Ax.  3) 


204  r*  -  396  r  =  285 
68  r^  -  132  r  =  95 


(Ax.  4) 


GM  =  4  X  17 ;  .-.  4  X  17  X  17  is  a  perfect  square  =  ^Ul 
17  X  m  r'  -  17  X  132  r  =  1615. 

(34  ry  -  (  )  4-  (33)-^  =  1615  +  1089. 
34  /•  -  ,\3  =  i  52 

..   .        2 "'^l*  _  1  ^5 — 

^       (229, 
.-.   a:  =  4:  2,  or  -t  V^^^iif*     ^^w. 

y=/-^  =  §X(i2)=4,5,and^i|V^^     J//.s-. 

5.    Given  (1)    3.r//  — 4.r  -  4// =  0  (2)  x*^  +  y^ +  ar  +  y - 
26  =  0 

(1,;    2xii-%{x^y)=^0 

(2i)  a:'^+y^4-(^-hy)  =  26 

x^  ^2xy  +  1/  -  ^{x-\-y)  =  2^         (Ax.  1) 

(^  +  //)''-j|(^  +  y)  =  -'6 

(^  +  yy  -  .«^  +  y)  +  {^y  ='26  +  il^  =  ».V 

(335j 


340  TEXT-BOOK    OF    ALGEBRA. 

(^  +  y)  -  ;}  =  i  V 

X  -{-  t/  =  (j,  or  —  ';•' 
a;  =  6  -  //,  or  -  \-'  —  1/ 
(1)  3(G-^)./  +  4y/-?4-4y  =  0  (228) 


yy2  _  0  y/  =  - 

8 

y-2  _  6^  +  <)  =  1 

//-o=il 

w 

y  =  4  or  2  j 

ir  =  2  or  4  \  A?is.                   (229) 

(1) 

^(-¥-y)y+¥+42/-4y/  =  0              (228) 

-  13  ij- 3  f -\-  %^  =0 

36  y'  +  12  X  13  n  =  208                        (334,  Ax.  3) 

36  7/^  +  ( )  +  (13)'-  =  169  4-  208  =  377 

6  y  +  13  =  -[-  V377 

-  13  -t  V377 

2/  =  — 6 

(4) 

-  13  =f  V377 

>Ans. 

1 

6              J 

(229) 

345.  Classes  of  Simultaneous  Equations,  two  or  more  being 
Quadratic,  and  the  others,  if  any,  simple  (344).  Methods  of 
solution. 

1.  Special  forms  akin  to  those  explained  in  342  can  be 
solved  by  a  similar  process.     See  Ex.  2,  344. 

2.  When  each  equation  contains  only  one  term  of  the 
second  degree,  and  that  term  has  the  same  product  or  square 
of  the  unknowns  in  all  of  the  equations.  See  Ex.  3  of  the 
preceding  article. 

3.  When  neither  of  two  given  equations  contains  the  first 
power  of  either  x  or  ?/.  This  is  usually  spoken  of  as  the 
homogeneous  case,  and  to  solve  it  y  is  put  equal  to  rx,  or  x 
equal  to  ry.     See  Ex.  4. 


QUADKATK      IJjl   A  I'loN.s.  341 

4.  When  an  equation  can  be  framed  which  can  be  solved 
tor  \x  function  of  the  unknowns.  See  Ex.  5  wliere  the  func- 
tion regarded  as  unknown  was  x  -\-  //. 

5.  Miscellaneous  methods.  —  Various  substitutions,  and 
expedients  of  other  kinds,  are  often  of  use  in  effecting  solu- 
tions in  i)ai'ticular  problems.  Suggestions  will  be  made  in 
the  proper  places  to  meet  such  cases. 

a.  The  methods  just  explained  are  not  mutually  exclusive.  Some- 
times an  exercise  can  be  solved  in  a  number  of  different  ways.  In 
some  instances  particular  methods  have  decided  advantages  over 
others. 

346.  Exercise  in  the  Solution  of  Simultaneous  Quadratic 
Equations,  two  or  more  of  which  are  of  the  second  degree. 

Jx'^-hy^=17()  (     a:^-f  3a-y  +  r  =  5 

"■'    \  xij  =  i;i  \2x^-\-xy-\-2  y"  =  5. 

(      3a;«-a-y-|-8y2  =  13.  (  8  x// =  336. 

•■l'(?+')-<v-4'r""-'"'X:""* 

'•    ^  .ry  -  V  =  .")().  ^'    \  v/2  -f  3  .r  -  4  //  =  2(). 


1 

J'  —       ■=  a 
// 


J  a;2  _  ^.,^  _|_  ^-2  =  3 


)  1        1  .1^-  I  ^2_2^y-f.4//-^  =  4. 


11. 


12. 


\2x^-■yxy^:^!r=\ 
\  3j-2— r)a-//  +  L>y  =  4. 


o 


J.2   _    3  ^y    -}_    2  ;/2   _    25     =   0. 


342  *  TEXT-BOOK    OF    ALGEBRA. 


14. 


15. 
16. 


4:Xf/  —  3  v/^  =  3  a. 

2i/  —  4:Xi/-\-3x''  =  17 


y^  —  x^  =  16. 
(  (1)  :r2  +  4  f  +  80  =  15  :r  H-  30  7/ 
I  (2)  xi/  =  6. 

Suggestion.  —  Multiply  (2)  by  4  and  add  it  to  (1).  Then  ar- 
range in  the  form  (x  +  2  yY  —  15  (x  +  2  2/)  =  —  56.  Solve  this  quad- 
ratic for  X  -\-  2y^  getting  two  values  for  it. 

j  x^  -\-  3  xij  =  54 
(  xij  +  4  f  =  115. 

Suggestion.  —  Add  the  two  equations  and  solve  for  x  +  2y. 

9  x''  +  //2  —  63  X-  -\-21y  -\-  86  =  0 
xi/  =  4. 

r  1  1       485 

x^~^y~W76 
11  _23 


17. 


18. 


19.  <^ 


1/ 


24 


20. 


Suggestion. — Solve  such  exercises  in  the  reciprocal  form,  i.e., 
without  clearing. 

'  x'^  ~\-  y  =  Ax 

y-2  -f  .T  =  4  //. 

Suggestion. —  Subtract  one  equation  from  the  other  and  divide 
through  by  x  —  y. 

r(i)   .^2  _^  ,/ -f  .^--^  =  30 

{{?y)x-y-r.  =  2. 

Suggestion.  —  Add  2  times  (2)  to  (1)  and    extract   the    square 
root  of  the  result. 


22. 


23. 


£C2  4-  y2  _j_   4    V^--^  +  If'  =  XI 

xy  =  12. 

j  £c2  -f  2  J-//  +  7/  +  3  x  =  73 


QUADKATIC   EQUATIONS.  343 


28. 


9A    \  '■  +  '^  -f  </  V.r  -I-  ?/  =  6  a* 

^'  I     y^  +  ^-^  =  b. 

f  (1)  X'  4-  1/  +  ^^  =  84 
25.  •{  (2)      X  -f  '/  +  ^  =  14 

Su(J(iESTioN.  —  Add  2xy  =  KJ  to  (1),  and  substitute  z  =  14  — 
(X  +  y)  in  the  new  equation,  and  after  arranging  solve  for  j  +  y. 

■  J  r  -f-  V^//  4-  //  =  l.*5. 
SUGGKSTiox.  —  Divide  the  first  equatioti  l)y  tlie  second. 

■|(2)  Vx9  =  i->. 

Suggestion. — Divide  (1)  through  l>y  Vx  —  Vy. 

U  a-*  -i-  s^  =  i 
[  y^  +  ^2  =  c. 

Suggestion.  —  Add  the  three  equations  and  divide  through  by  2. 
From  this  equation  subtract  each  of  the  equations  in  turn. 

^   [  X  =  fl  Var  +  y 

Suggestion.  —  Square  each  equation  and  tluMi  sul)traet  one  fr(Mn 
the  otlier.  Tlu*  resulting  etiuation  is  divisible  by  .r  +  //.  Also  divide 
one  of  tb«*  «M|U!iti()ns  by  tli«*  oiIht. 


SECTION    III.  —  Problems. 

347.    Problems  Involving  the  Solution  of  Simultaneous  Quad- 
ratic Equations. 

1.  Find  two   numbers  wliicli    nuiltii>li«'(l    ^ivc    nTd   and 
divided  the  one  by  the  other  give  2\. 

2.  Two  nnml)ei-s  are  to  eacli  other  as  11  to  1,'^,  and  the 
sum  of  tlicir  sfpiarps  i<  14210.     Wlint  are  tlif  niinil»Ts? 


344  TEXT-BOOK  OF  ALGEBRA. 

3.  Find  two  numbers  whose  sum,  product,  and  difference 
of  their  squares  are  all  equal  to  each  other. 

4.  What  number  being  divided  by  the  product  of  its 
digits  gives  the  quotient  2,  and  if  27  be  added  to  the 
number  the  digits  will  be  inverted  ? 

5.  What  numbers  are  there  whose  sum  is  100  and  the  sum 
of  whose  square  roots  is  14  ? 

6.  A  and  B  have  each  a  small  field  in  the  shape  of  a 
square,  and  it  requires  200  rods  of  fence  to  enclose  both. 
The  sum  of  the  contents  of  these  fields  is  1300  sq.  rods. 
What  is  the  value  of  each  at  $2.25  a  square  rod  ? 

7.  AVhat  two  numbers  are  those  whose  sum  multiplied  by 
the  greater  is  120,  and  whose  difference  multiplied  by  the 
less  is  16?  • 

8.  Two  farmers  together  drove  to  market  100  sheep,  and 
returned  with  equal  sums.  If  each  of  them  had  sold  his 
sheep  at  the  price  the  other  actually  did,  the  one  would  have 
returned  with  $180  and  the  other  Avith  $80.  At  what  price 
per  sheep  did  they  sell  respectively  and  how  many  sheej) 
had  each  ? 

9.  The  small  wheel  of  an  ordinary  bicycle  makes  135  revo- 
lutions more  than  the  large  wheel  in  a  distance  of  260  yds. ; 
if  the  circumference  of  each  were  one  foot  more  the  small 
wheel  would  make  27  revolutions  more  than  the  large  wheel 
in  a  distance  of  70  yds.  Find  the  circumference  of  each 
wheel. 

10.  A  tailor  has  noticed  that  broadcloth  on  being  wet 
shrinks  up  ^  in  its  length  and  ^\  in  its  breadth.  If  the 
surface  of  a  piece  of  broadcloth  is  5^  square  yds.  less,  and 
the  distance  round  it  4|  yds.  less  than  before  it  was  wet. 
what  was  the  length  and  width  of  the  broadcloth  originally  ? 

11.  A  cistern  which  is  half  full  can  be  filled  by  one  of 
two  pi])es  in  a  certain  time  and  eni])tied  by  another  in  a  dif- 


QUADKATK      i:(,)r  ATIONS.  345 

ferent  time.  If  both  pipes  be  left  open,  the  cistern  is  emp- 
tied in  12  hours.  But  now  if  the  opening  of  both  pipes  be 
macle  smaller  so  that  the  one  needs  an  hour  longer  in  tilling 
and  the  other  also  an  additional  hour  in  emi)tying,  then  if 
both  pipes  are  left  oi)en,  15^  hours  are  needed  to  empty  the 
cistern.  In  what  time  can  the  empty  cistern  be  filled  by 
the  first  pipe  alone,  and  in  what  time  can  the  full  cistern 
be  emptied  by  the  other  acting  alone  ? 

12.  A  certain  number  of  laborers  remove  a  heap  of  stones 
from  one  \)\aee  to  another  in  8  hours.  Were  the  number  of 
laborers  8  more,  and  did  each  carry  2]^  kilograms  less  at  a 
time,  then  the  heap  would  be  transported  in  7  hours.  But 
if  tlie  number  of  laborers  were  8  less,  each  carrying  5 J  kilo- 
grams more,  then  the  heap  would  be  removed  in  9  hours. 
How  many  laborers  were  there,  and  how  many  kilos  did 
each  carry  at  a  time  ? 

13.  Find  two  numl)ers  whose  sum  added  to  the  sum  of 
their  squares  is  42,  and  whose  product  is  15. 

14.  Find  two  numl^ers  such  that  their  product  added  to 
their  sum  shall  l)e  47,  and  their  sum  taken  from  the  sum  of 
their  squares  shall  leave  62. 

15.  Find  three  numbers  such  that  when  the  sum  of  the 
first  and  second  is  multiplied  by  the  third,  the  product  is 
63;  when  the  sum  of  the  second  and  third  is  multiplied  by 
the  first,  the  product  is  28 ;  and  when  the  sum  of  the  third 
and  fii-st  is  multiplied  by  the  second,  the  product  is  5o. 

Sl'goestiox.  —  Solve  in  a  manner  similar  to  Ex.  28,  346. 

16.  The  diagonal  of  a  lx)x  is  125  inches,  the  area  of  the 
lid  is  45(M)  sq.  in.,  and  the  sum  of  the  conterminous  edges 
is  215  inches.     Find  the  lengths  of  these  edges. 


346  TEXT-BOOK  OF  ALGEBKA. 


CHAPTER   XXTII. 

QUADRATIC   EQUATIONS  AND   EQUATIONS   IN   GENERAL. 

348.  Properties  and  Solutions  of  Quadratic  Equations  and 
Equations  of  Higher  Degrees.  Some  of  the  properties  of 
quadratic  equations  obviously  belong  to  equations  of  other 
degrees.  It  will  be  found  also  that  many  equations  of 
other  degrees  can  be  solved  like  quadratics.  It  will  be  con- 
venient besides  to  introduce  into  this  chapter  the  general 
discussion  of  problems  and  the  validity  of  processes  of 
solution. 

SECTION  I. 

Properties. 

349.  The  Sum  of  the  Roots  of  a  Quadratic  Equation.  — 

When  the  coefficient  of  x^  is  unity,  the  coefficient  of  x 
taken  negatively  is  equal  to  the  sum  of  the  roots. 

If  in  the  type  form  of  quadratic  equation,  ax'^  -}-  2  bx  ==  c, 
the  right  member  be  transposed,  and  the  equation  divided 

b  c 

through  by  a,  there  results  x^  —  2  -  x  —  ~  =  0.    To  simplify 

b  c 

the  equation  we  replace  —  2  ~hy  p,  and  —  -  by  q,  and  let 

us  call  the  new  ecpiation  the  normal  for7n. 
x'^  -\-  px -\-  q  =0 
4a;2  +  4^^^  =  _  4  ^  (334) 

{2xy  +  (  )  +  ;>2  _  ^2  _  ^ (335) 

2  £c  -f  ;;  =  _j_  Vi>^  —  4  ^ 

^=-hp-\-\  V>  '  —  4  y 
OT  X  =  —  ^j)  —  \  yj  p'^  —  4  5' 


QUADRATIC   ?:QUATIONS.  347 

If  now  the  two  values  of  a*,  which  are  the  roots  of  the 
e() nation,  be  added,  the  radical  parts  cancel  and  the  sum  is 
—  yj,  i.e.,  the  sum  of  the  roots  is  equal  to  the  coefficient  of  x 
in  the  normal  equation  taken  with  opposite  sif/n.     Q.  E.  D. 

350  The  Product  of  tbe  Roots  of  a  Quadratic  Equation.  — 
The  product  of  the  roots  is  equal  to  the  known  term  in  the 
normal  equation. 

Multi])lying  together  the  two  roots  of  the  last  article, 


{;-^v/. 


4  -  \  0''  -  4  7)  =  '/    Q  K.  D. 
4        4 


351.  Solving  a  Quadratic  Equation  by  Factoring.  — By  com- 
paring the  theorems  of  the  last  two  articles  with  116,  8,  we 
learn  tAxat  fartorifuj  the  normal  form  of  any  quadratic  equa- 
tion gives  its  roots. 

1.  Given  the  ecjuation  x-  -f-  7  ar-  4-  1-  =  <>  to  Hud  its  roots. 
Ftictoring  the  left  member. 

(X  4-  4)  (.r  +  :\)  =  0.  (135,  2) 

By  349  and  350  the  roots  are  —  4  and  —?>.  For,  (—  4) 
-(-  (—  .S)  =  —  7  which  is  the  coetficient  of  x  taken  neya- 
tirt'h/ :  while  —4  X  —  ''^  =  +  12,  which  is  the  known  term. 
Hence,  after  factoring,  tiike  the  second  terms  of  the  factors 
with  their  siijns  rhaufp'd  for  the  roots  of  the  ecjuation. 

2.  x'—Q,  -\-h)x  -!-///>  =  () 

(:r  —  a)(x  -  h)  =0.  (135,2) 

li«Mice  the  roots  are  a  and  //. 

XoTK.  —  Compare  this  solution  with  that  <;iv<'ii  l>v  the  rrmilar 
process,  Ex.  32,  338 

3.  Solve  x' —  7  r -f- rj  =  0  by  iiicloriug,  and  al.s«»  Uy 
completing  the  square. 


348  TEXT-BOOK    OF    ALGEBRA. 

4.  x^-9x-\-20  =  0.  9.  :c2_20a: -300=0. 

5.  ^2  _^  11  ^  4-  30  =  0.  10.  x'  -Sx  =  —  15. 

6.  a;2_^13a;  +  12  =0.  11.  x' -  llx  -  50  =  160. 

7.  ^2  +  21  ^»^  4- 110 ^2  =  0.  12.  x^- |a.T  +  f4  =0. 

8.  a-2  —  G  a;  —  27  =  0.  13.  6  a^^  _  17  ^  ^  jL2  =  0. 

Remark.  —  Factoring  as  in  135,  3  we  get  (3  a;  —  4)  (2  x  —  3)  =  0. 
Or  (x  —  |)  (x  —  |)  =  0  by  dividing  the  equation  through  by  3  X  2 
(Ax.  4.)     Hence  the  roots  are  4  and  |. 

14.  a •  +  4  +  ^-  ^^^  =  13        17.    6  ic2  _  ^  _  12  =  0. 

X 

15.  ^ili  + ^^H--«  =  7        18.    ^2_o3^^o.7. 

X  x^ 

16.  ^+3i=y^  +  8. 

352.    Solving  Equations  of  Any  Degree  by  Factoring. 

1.  Let  us  verify  some  of  the  examples  of  the  last  article, 
substituting  the  values  not  in  the  original  but  in  the 
factored  form. 

Putting  X  =  —  4  in  Ex.  1,  we  have 
(-4  +  4)  (-4  +  3)  =0,  or,  0  (- 1)  =  0.  (Ill,  a) 

Thus  when  one  factor  of  a  product  is  zero  the  product  is 
zero,  and  so  the  equation  is  verified.  Substituting  x  =  — 
3  will  make  the  second  factor  zero  and  so  satisfy  the 
equation.     (189,  a.) 

Likewise  in  the  second  example,  either  x  —  a,  ov  x  =  h, 
will  make  one  of  the  factors  zero,  and  so  satisfy  the  equa- 
tion, and  they  are  therefore  roots  of  the  equation. 

2.  But  this  reasoning  will  hold  for  any  number  of  factors 
as  well  as  for  two.  Thus  the  equation  (x  —  5)  (x  -\-  2)  (x 
—  1)  =  0  (oi^,  x^  —  ^  x^  —  1  X  -\-  10  =  0,  by  multiplication) 
is  satisfied  by  either  .r  —  5  =  0,  a*  -f-  2  =  0,  or  a^  —  1  =  0. 

.-.  ./'  =  5,  a*  =  —  2,  X  =  1. 

And  these  are  the  three  roots  of  the  equation. 


(,»l   AI)1;ATI<'    K(,>rATInNS.  349 

Again,  taking  the  tujiuition  0  x*-  -j-  5  u:  —  4  =  0,  we  may 
write    it   in   the    form    CJ  ./•  -  1)  (.S  ^  4- 4)  =  0.      (135,3). 

Placing  each  factor  in  tmn  ripial  \n  /cid. 

'J  J-  —  \  ={)  .-.  ./•  =  1.  :  an. I  ."I  .<•  -}-  4  =  0  .-.  ./•  =  —  * 

3.  When  the  unknown,  a*,  divides  out  of  an  equation,  one 
root  of  that  equation  is  zero.     Thus  in  the  equation 

2x^-nx'  -\-  V2x^x{x-4)  C2x-'A)  =(), 
placing  each  factor  ecjual  to  zero, 
^  =  0 :  r  —  4  =  (I.  .-.  ./•  =  4  ;  2  J-  —  .'J  =  i).  .'.  x  =  [\. 

Manifestly  r  =  0   satisfies   the   ec^uation   2x^  —  li  x'^ -\- 

1- r  =  ().  foi-  it  makes  every  t«'rm  (Mjual  to  zero. 

353.  Rule  for  Solving  Equations  of  Any  Degree  when  they 
can  be  Factored. 

1.  15y  ti-ans position  if  necessary  make  the  right  member 
zero. 

2.  Factor  the  left  meml>er  into  binomials  the  first  term 
of  each  of  which  is  a  multiple  of  the  unknown. 

3.  Set  each  fiu-tor  equal  to  zero,  thus  obtaining  as  many 
values  of  the  unknown  as  there  are  binominal  factors. 

The'  following  corollaries  of  this  method  are  highly  im- 
|K)rtant  in  the  genei-al  theory  of  equations.  Their  truth  is 
merely  suggested  here. 

a.  An  equation  has  as  many  roots  as  then'  are  units  in  its  <ie{;ree. 

b.  If  ac  —  n  is  one  factor  of  the  left  member  of  an  e<] nation  >yl»ose 
right  member  is  zero,  then  a  in  a  root.  Hence  if  we  can  find  one  or 
more  factors  wlicthir  we  can  find  the  others  or  not,  tlicsc  factors 
give  roots. 

c.  If  one  or  mun-  untoi-s  an*  known,  the  equation  divided  by  these 
leaves  an  equation  which  contains  the  other  factors,  or  the  other 
roots. 


350  TEXT-BOOK    OF    ALGEBRA. 

354.  Exercise  in  the  Solution  of  Equations  by  Factoring, 

1.  x~  -f  23  ax  -f-  90  a'  =  0. 

2.  G  x'-"  +  17  X-  -f  12  X  =  0.  •     • 

3.  V2x^  —  9  X-  —  8  it-  -[-  6  =  0. 

4.  cf3  4-  a;2  +  a;  +  1  =  0. 

5.  ic  (:z;  -  6y  +  10  a:  —  60  =  0. 

Suggestion.  —  This  equation  is  evidently  divisible  by  x  —  6  giving 
a  quotient  x-  —  6  x  +  10  =  0.  Hence  G  is  one  root  and  the  others 
must  be  found  from  the  equation  x-  —  G  r  +  10  =  0.  It  is  not  pos- 
sible, however,  to  resolve  this  equation  by  factoring.  Turning  then 
to  the  old  process  of  completing  the  square  the  roots  are  readily 
found  to  be  X  =  3  ±  V  —  1.  So  in  all  cases.  If  an  equation  can  be 
reduced  down  to  the  second  degree,  the  solution  can  always  be 
perfected  by  the  old  process. 

6.  x^  +  Ax'  —  ix  —  16  =  0. 

7.  x^  —  2x^  —  16  x^  +  32  X  =  0. 
S.    x^  —  31  x'  +  S^x  -1  =  0. 

9.    x^  —  5  x^  -  6x2  ^  15  ^.  _^  9  ==  0. 
Suggestion.  —  Arrange  into  x^  —  G  x-  +  9  —  5  x  (x^  —  3)  =  0. 

10.  a;*  -f  7  ax^  -  2  a^  x^  —  2S  a^  x  —  8  a\ 
Suggestion.  —  Arrange  as  follows :  — 

X*  -  2a2  x2  -  8  a*  +  7  ax  (x^  -  4  «2)  =  o. 

11.  One  root  of  the  equation  x^  —  3x'^  —  14:  x  -\-  42  =  0 
is  3  ;  find  the  other  two. 

12.  Two  roots  of  the  equation  x^  —  2  x^  -{- 1  x'^  -\-  2 'x  ~ 
5  =  0  are  1  and  —  1  :  find  the  other  two. 

13.  Solve  the  equations  27  x^  —  1  =  0 ;  x^  -\-  1  =0. 

355.  Conversely.  —  To  Construct  an  Equation  when  its  Roots 
are  given :  subtract  each  root  from  x  and  iuulti])ly  all  the 
resulting  binomial  factors  together. 

((.  Thus  far  we  have  always  found  irrational  and  imaginary  roots 
occurring  in  pairs,  one  with  the  —  sign  before  its  radical  term  and 


(,haih:a  ric   i.(,»i  a  rioNs.  851 

the  other  with  the  +  sij^ii  before  its  mtlical  t«'nn.     They  are  some- 
times deseril>e(l  as  eonjiijjate  surds  or  imaginaries. 

Uixm  multiplying  two  sinli  liinoniial  factors  a  trinomial  with  real 
and  rational  coetticients  results,  i'hus  taking  the  two  imaginai7 
roots  of  example  5  of  the  preceding  article, 

a;  -  (3  +  V  -1) 

g  -  (8  -  V  -  1) 

x2  -  3  X  -  ar  V^^ 

„  Z  3  X  -f-  a  V^^  4-  9  -  (  -- 1) 

x*-6i  +  10. 

Furthermore  it  is  easy  to  see  that  if  the  radical  terras  did  not  dis- 
appear from  this  proiluct,  they  would  certainly  not  disappear  when 
multiplied  by  dissimilar  radicals  in  the  other  roots,  and  so  would  not 
disappear  in  the  final  product,  which  is  the  equation.  Ilejice  we 
conclude  that  if  an  equation  liaviiii:  real  coetticients  does  have  imag- 
inary or  irrational  roots  they  (><( m  in  pairs,  and  should  be  taken 
toyether  in  reconstructing  the  e<mation. 

1.  Form  the  equation  whose  roots  are  -j-  3  and  —  4. 
(x-3)(x  -(-  4))  =  0,  or  x^  +  a:  -  12  =  0. 

2.  Foriu  an  equation  whose  roots  are  1,  2,  an«l  —  3. 

3.  Construct  the  equation  whose  roots  are  0,  —  1,  2, 
and  —  5. 

4.  Find  the  equation  whose  r<)<it-    iiv  7  -f  v  .J.  7  —  V3, 
and  1. 

5.  What  is  the  equatmn  whox-  roots  are,  1  ^  V2,  2  ^^ 
V-  3,  i.e.,  1  -f  y2,  1  -  V2.  'J  +  \  ^^  2  -  V^^. 

6.  Form  the  ecjuation  with  the  roots  2  ^  V^,  —  2.  and  1. 


352  TEXT-BOOK    OF   ALGEBRA. 


SECTION    11. 

Discussion  of  Pkoblems.  —  Eauakk  of  Pkocesses  of 
sot.utiox. 

356.    Discussion  of  the  Equation  of  the  Second  Degree. 

Greatest  and  least  values  of  coefficients. 
Let  us  take  the  equation  as  found  in  349, 


the  roots  of  which  are  x  =  —  |  -|-  I  v/>""  —  iq. 

There  are  two  classes,  real  roots  and  imaginary  roots. 

1.    Equations  having  real  roots. 

In  order  that  the  roots  may  be  real  the  radical  quantity 
■y/p'^  —  4:q  must  be  zero  or  positive. 

(1.    li p^  —  4y  =  0,  the  radical  in  the  roots  disappears, 

and  X  =  —  i^>,  or  X  =  —  Ip,  i.  e.,  the  two  roots  are  equal. 

7>^        I         P 
In  this  case,  x^  -^  px  -\-  q  ^=i  x^  -\-  px  -\-  jr-^^   (  •^'  +  2 


X  +  '^  )  =0,  thus  giving  two  roots  each   equal  to  —  \-- 

(351) 

(2.  If  p'^  —  \q'>  0,  the  roots  are  real.  Let  us  suppose 
a  and  h  to  be  the  two  roots.  The  signs  of  the  roots  depend 
upon  the  signs  of  p  and  q.  Compare  articles  116,  8,  135,  2, 
and  351. 

(1)  x'-\-yx  J^q^ix-^  a)  (x  +  ^)=  0, 

or,  when  2^  and  q  are  both  positive,  the  roots  are  both  nega- 
tive. 

(2)  x^  -px-\-q  ^  {x  -  a)  (x  -b)  =0, 

i.e.,  when  p  is   negative  and  q  positive,  the  roots  are  both 
positive. 


yL'Ai>i;  \  Tie   k<m'  xrioNS.  353 

(3)  a;2  _|_^^a-  —  q^  {X  —a)  {x  +  0)  =  (►,  wheiv  a  <  ^, 
i.e.,  lip  is  positive  and  j'  negative,  the  less  root  is  positive 
and  the  greater  is  negative. 

(4)  x^  —  pjr  —  y  —  (j-  -+-  a)  (x  —  b)  =  0,  where  a  <  h, 
i.e.,  if  p  and  (j  are  both  negative,  the  less  root  is  negative 
and  the  other  positive. 

2.  Equations  having  imaginary  roots.  Here  the  quantity 
p^  —  4q  under  the  radical  sign  is  negative.  If  we  conceive 
either  p  or  q  to  vary  in  value,  say  p,  the  quantity  p^  —  4^ 
will  also  change,  and  may  pass  from  -f  to  — ,  or  from  —  to 
-f.  But  in  making  such  a  change  at  one  epoch  it  must 
have  the  value  zero.  Corresponding  to  this  is  the  critical 
value  of  X :  for,  as  long  as  the  quantity  under  the  radical 
sign  is  positive  the  value  of  the  unknown  is  real,  and  as 
soon  as  it  becomes  negative  the  value  of  the  unknown  be- 
comes imayinari/.  The  value  of  q  must  be  positive  to  give 
imaginary  roots :  for,  so  long  as  q  remains  negative  p'^  —  4y 
can  never  be  anything  else  than  positive,  making  Vp*  —  iq 
real. 

To  illustrate.  —  Suppose  2^  =  4,  then/?=^  _  4  //  =  0  gives  p 
=  ^4:^  and  80  p  cannot  be  greater  than  4  or  less  than  —  4 
in  the  equation  x*  +  /?j;  +  4  =  0 ;  for  the  moment  that  it  is 
made  larger,  -s/p"^  —  Aq  becomes  imaginary,  i.e.,  x  becomes 
imaginary.  Such  values  of  p  may  be  called  maximum  or 
minimum  values,  because  they  are  the  greatest  or  least 
which  yield  real  values  of  .r. 

357.  *  Manilla  and  Minima  Values  of  Functions.  —  It  //  be 
a  fun(*tion  of  a?,  and  if  as  x  increases  y  increases  for  a  time 
and  then  decreases,  the  greatest  value  that  y  attains  is  a 
ma.ximum ;  but  if  as  a;  increases  y  decreases  for  a  time  and 
then  increases  the  least  value  that  y  attains  is  a  minimum. 

n.  Functions  are  of  two  kinds,  explicit  and  implicit.  Thus,  in  y 
=  x'-*  +  2x  +  5,  y  is  an  explicit  function  of  x,  its  value  being  given 
directly:  while  in  x^  +  xy  =  //-  +  :5  y  -f-7,  y  is  an  implicit  function  of 


354  TEXT-BOOK   OF   ALGEBRA. 

X,  since  to  find  its  value  in  terms  of  x  it  is  necessary  to  solve  the  equa- 
tion for  y.     This  gives 


x-S   ,    1 


y  =  —^ —  ±  i^  V-19  —  6  X  +  5  X-' 
and  y  has  now  become  an  explicit  function  of  x. 

1.    Given  2/  =  ^•"  +  6  x  —  17  to  find  the  minimum  value 

To  show  that  there  is  a  minimum  value  of  //,  we  substi- 
tute a  series  of  values  of  x  in  the  equation  and  hnd  the  cor- 
responding values  of  y. 

Kow  x  =  —  9,  or  any  larger  negative  number,  makes  y  — 
10,  or  some  increasingly  larger  positive  number. 

( When  X  =  —S   x  =  —   7   x  =  —    6   x  =  —    5   x=  —   4 
(then      y=-V  y  =-10'  y= -17'' y  = -22' y= -25' 

x  =  —    3     x=  —    2     x  =  —    1     X  =        0 
?/  =  -  26  '  y=—2o'  y  =  —22'  y=  —17' 

X  =         1     X  =  -{-2 
y=-10'  y  =  -l' 

and  «  =  3  or  any  larger  positive  number  makes  y  =  -\-  10 
or  some  increasingly  larger  number. 

By  examining  we  see  that  as  x  changed  y  reached  its 
least  value,  —  26,  when  x  was  —  3.  To  find  the  exact  limit 
we  proceed  as  follows  :  tve  solve  the  equation  for  x,  and  then 
see  for  what  value  ofy  the  radical  becomes  iina^ilnanj. 

y  =  x'^  -^  6x  —  17 

x^  -\-  6x  =  y  -\-  17 
ic2  _^  6  X  -h  9  =  ?/ +  26 
ic  -f  3  =  i  V//  4-  26 
x=  -3  ^-Vy-\-26. 

The  radical  passes  from  real  to  imaginary  just  before  it 
changes  to  a  minus  quantity,  i.e.,  at  zero.  Hence  to  lind- 
the  limiting  value  of  y,  we  place 

7/+26  =  0 
Whence,    y  =  —  26. 
This  value  of  y  is  the  minimum  value. 


(,>l   .\I>1:A  IK       I  <,»l   .\  1  1(»NS.  355 

2.  Given  y  as  tlie  implicit  lunctioii  of  x  as  contained  in 
the  e<iuation  y^ -}- u*-^  =,*).»• -f- 4 // to  tiiid  the  uiMximinii  or 
niininiuni  values. 

l>v  substituting  we  Icaiii  that  then*  is  l)otli  a  niaxiimiiii 
and  a  niiniunini. 

11  .r  ==  —  '2  or  any  greater  negative  number,  y  is  imagi- 
nary. 
^lfa-  =  —  1  j-  =  0  x-\  x-'l 

then  >/ =  \        +11/=     lO.     2/=     i—    Ah     y  =     \—    .4.> 


,  (  /or, +  2'  (4.'  (  +  4.4.-)'  j  +  4.4;"> 

a-  n  :j         J.  =  4 

\r  i  +  25 

and  ic  =  5  or  any  greater  posit  i\f  immbtM-.  y  is  imaginary. 
To  find  the  exact  value 

,/-  :{-  a:^  =  .'{ •'•  +  4  y 

X'  -  3.r  +  5  =4y~yH-  5  (335) 

^  -  i  =  i  V^  +  4  //  -  //^ 
Then  ?  -|-  4  //  —  //-  =  0  (See  previous  exam})le) 

//•-  —  4  //  =  ; 
//•^  -  4  //  +  4  =  V 

y  ^  4.5  or  —  .5. 

Returning  to  the  series  of  values  given  above  we  see  that 
4.5  is  the  maximum  vahu^  between  4.4.")  and  4.45  (the  func- 
tion was  increasing  at  the  tii-t  ainl  d'  •  i. -ising  at  the  second, 
so  that  it  must  have  reached  a  maximum  at  scnne  point 
l)etween),  Avhile  —  .5  is  the  minimum  between  the  values 
—  .45  and  —  .45. 

Substitutini.c  (229),  the  ('(HTcspoiKliiig  value  of  ./•  is  lomid 
to  1k'  1.5. 

These  values  may  be  inserted  in  the  above  series, 

a-  =  l  r=        1.5  a;  =  2 

Thus,  (  -  .4.-)  (  -.5  (  -  .45 

^=    (      I.4.V'/=  \     \.ryy=   \     4.45 


356 


TEXT-BOOK   OF    ALGEBRA. 


3.    Divide  a  given  number  a  into  two  parts    such   that 
their  product  shall  have  the  greatest  possible  value. 

Let  X  =  one  part 

a  —  X  =  the  other  part 
then  X  (a  —  ic)  =  ?/,  is  a  maximum. 
x^  —  ax  =  —  y 

cr        (t^ 

-y 


X- 

-  «-^-  +  4  =  T 

X 

8o  that  for  the  maximum ?/  =  0  whence  y  =  —. 

Corresponding  to  this,  x  =  |,  i.e.,  the  maximum  product 
is  obtained  when  the  two  halves  of  the  number  are  multiplied. 

4.  To  inscribe  in  a  triangle  a  rectangle  of  maximum  area. 
There  is  such  a  rectangle ;  for  if  we  conceive  of  a  rec- 
tangle at  the  bottom  of 
the  figure  having  a  base 
nearly  as  long  as  the  base 
of  the  triangle,  BC,  (and 
consequently  a  very  small 
altitude)  have  its  base 
diminish  in  size  and  its 
altitude  increase,  it  will 
finally  come  to  have  an 
altitude  nearly  equal  to  the  altitude  of  the  triangle,  but 
with  a  very  small  base.  Evidently  if  the  rectangle  first  in- 
crease in  size  and  then  diminish,  there  must  be  some  posi- 
tion for  which  it  has  a  maximum  area. 

By  a  theorem  in  geometry,  since  the  triangles  ABC  and 
ADE  are  similar, 

AP  :  AQ  : :  BC  :  DE  : 
Call  BC,  h ;  AP,  h ;  and  y,  the  area  of  the  rectangle. 
Required  to  find  the  maximum  value  of  the  last. 


(.•lAhUA  I'M       l.f^tr  A  rioNs. 

Substituting  in  the  precedm--  proixtitioii, 
h:h~.r::   h :    \)K. 

// 
unci  since  the  area  of  a  reetuni^U*  =  imse  x  Jiltitnde. 


V  - 


h 

hij  =  hhv  -  hx-  (Ax.  3) 

\lSij,i  _  4  h^hx  =  -  4  hlnj  (334) 

4 yijc'^  _  4  U'hx  +  Irh^  =  h'h'-  -  4  bhf/ 
1>  f,x  —  bh  =  i  VV>'//-  -  4  hinj. 

Now  i*/('^  —  4  bhif  =  0  t^'ivcs  tlu'  iiiaxinuun. 

Hence  2  />.r  —  hh  =  0  ;  or  ./•  =  !;  :  //-A-  —  4  b/t  i/  ^  0  :  or  // 


Tluis  the  niaxiniuni  rectangle  lias  for  its  altitude  ha/f  the 
altitude  of  the  triangle,  and  for  its  area  ^  of  the  btise  X  the 
altitude,  which  is  half  the  area  of  the  whole  triangle. 

5.    Determine  the  rainimuni  and  maximum  values  of  the 

function  y  =  ^—^ — ^  ti"    *^^  ^®^^  values  of  x. 
•^  (>:r  —  14 

a 
358.    Discussion  of  the  Equation  x  =  t-  for  Limiting  Values 

of  '/  and  //.    Infinites  and  Infinitesimals. 

a.  As  stated  in  62,  the  symbol  0  is  often  used  to  denote  any  ex- 
ceedingly small  number  less  than  any  that  can  lie  named.  Sueh 
quantities  are  calle<l  infinitesimals.  Any  number  greater  than  any 
that  can  be  named  is  called  an  infinite.  A  finite  quantity  is  neitluT 
an  infinite  nor  an  infinitesimal.  The  student  nnist  undei-stand  at 
the  outset  that  all  infinitesimals  are  not  of  the  same  size,  nor  are  all 
infinites  of  the  same  size,  though  0  is  used  to  denote  all  of  the 
former,  and  »  all  of  the  latter. 

b.  The  sign  =  (  read  "approaches  the  limit")  is  used  to  signify 
that  one  quantity  approximates  to  the  value  of  another  more  and 
more,  so  that  they  differ  by  less  than  any  assignable  magnitude. 


:]>')><  TKXT-r.OOIv    OF    Al.CKHP.A. 

1.    To  find  the  value  of  y  when  the  denominjitor  h  is  finite^ 

but  the  numerator  is  infinitesimah 

The  result  may  be  arrived  at  perhaps  better  by  an  exam- 
ple than  in  any  other  Avay.  Suppose  b  is  any  number,  say 
500,  and  a  assumes  the  values  100000,  10000,  1000.  100,  10, 
1,  .1,  .01,  and  so  on  without  end,- 


500 ^'^y     500  -^^h    3^0  0    — -^  ?    5  00  —  5? 

JJL  =  J„     -  1„_  =  -1^     _l_    =  _J _^(LL  —  _^1 

5  0  0  5  0?    5  0  0  5  0  05500  500055  0  0  500  0  0? 


and  so  on  without  end.  Evidently  the  value  of  the  fraction 
gets  smaller  and  smaller,  or  what  is  the  same  thing  ap- 
proaches the  limit  zero.     Stating  this  ])rinciple  generally,  if 

in  the  fraction  j,  the  numerator  approach  the  limit  0,  the 

fraction  approaches  the  limit  0.  Stated  in  the  form  of  an 
equation : 

a 


(1)    AVhen  a  -j=  0,  x  =  7=0. 


2.  Similarly  the  value  of  j.  wlien  (/  x^i,  finite  and  i>  iiifi iiit<\ 
a])])r();icli(^s  the  limit  zero. 

3.  To  find  the  value  of  j  when  a  is  finite  and  b  infinites- 
imal. 

This  is  the  reverse  of  the  first  case.  AVhatever  finite 
value  the  numerator  may  have,  as  the  denominator  dimin- 
ishes the  value  of  the  fraction  increases,  and  ultimately 
when  the  denominator  has  become  infinitesimally  small,  the 
value  of  the  fraction  has  grown  infinitely  large. 

(2)    WhenZ»:^0,  ^^x. 

4.  AVhen  b  is  finite  and  a  is  infinite  the  fraction  ap- 
proaches the  same  limit,  cc  . 


(  M'AI>l:  \  !  1<       1  •  'I    A  11.  "NS.  359 


5.  lo  find  tli»'  valiu'  (»t  ./•  ==  when  lx)th  a  and  h  are 
inhnitesiinals. 

In  the  first  exuniph*  given,  how  t'\ ci'  small  the  (U'noiniiialor 
might  become,  the  value  d  iIm-  traction  wtis  always  5^^^  of 
it.  Thus  of  these  two  infinitesimals  one  was  500  times 
greater  than  the  other.  In  general  when  h  is  finite  and  a 
infinitesimal  the  value  of  the  fraction  is  an  infinitesimal 

which  is  -J  times  the   first.     Now,  if  two  quantities  grow 

small  together  and  we  have  no  means  of  knowing  their  rel- 
ative size,  ihe  mJin'  of  the  frurfltni  Is  huli'frrttnntifp, 

(3)    AVhcn  (I    '    o.  and  A    "    »».  ./  is  ui(h't(>rw'niate. 

h 

However,  in  such  a  fiat  lii mi   i  ^         ,  \i  x  be  infinitesimal, 

the  value  of  the  fraction   does   not  become  indeterminate, 
since  the  some  infinitesimal  is  in  l)oth  terms  and  so  divides 

fl2  _j_  i/t 
out.  giving  T).     Likewise  the  fraction j- does  not   be- 


(M»m<'  1 


ndeterminate  when 


.         <i'-  —  fr        (a  —  b)  {a  -\- h) 
it  —  b  (a  —  b)  X  1 

359.  *  Problem  of  the  Lights.  —  ihis  celebrated  problem, 
due  to  Chiiraut,  furnishes  an  cxt  (  llciit  exercise  in  the  inter- 
]»retation  of  algebi-aic  results. 

A m — ^1-^:,-* B 

p  '     !        -^    Q 

Two  lights  are  at  P  and  Q.  It  is  required  to  find  the 
points  on  A  B  having  equal  illumination. 

By  a  law  of  optics  the  intensity  with  which  a  light  shines 
at  any  outside  point  is  inversely  proportional  to  the  square 
of  the  distance  from  tlie  ])oint  to  tlie  source  of  the  light. 


360  TEXT-iiooK  OF  al(;i:p.ka. 

Let  a  =  PQ  be  the  distance  between  the  lights, 

nri^  =  the  intensity  of  illumination  of  the  light  P  at 

the  distance  of  one  ft.  from  the  source, 
n^  =  the  similar  intensity  of  the  light  Q. 

Suppose  I  to  be  the  point  of  equal  illumination ;  and  let 
X  =  PI,  then  Ql  =  a  —  x.     If  now  the  illumination  from  P 

111 

is  m^,  at  1  ft.,  it  is  ^  m^  at  2  ft.,  -  m^,  at  3  ft.,  --z  m\  at  x  ft. 

4  9  x-^ 

according  to  the  law  of  intensity.    Likewise  the  illumination 

from  Q  at  a  distance  of  a  —  x  ft.  is  ~, r-^.     But  by  the 

(a  —  x)'^  ^ 

supposition  the  two  intensities  are  equal. 


m- 

n' 

x^ 

-  (>/  -  xf 

m 

n 

X 

-^a-x 

ma  —  ')nx 

—  -1-  nx 

(m  :^  n)  X 

=  ma 

ma 

(Ax.  6) 


m  m 

^  —        ,       I.e.  j —  a,  or a. 

m  -J-  w  7n-\-  11  r/i  —  7i 

From  the  v^alues  of  cc,  we  learn 

1.  That  there  are  two  points  equally  illuminated. 

2.  One  position  of  the  point  of  illumination  may  always 
be  found  ivithin  the  line  PQ.     For,  since  by  the  nature  of 

the  problem  m  and  n  can  only  be  positive  numbers — , — 

■^         -^  7n-\-n 

will  always  be  a  proper  fraction,  and  therefore  ;     ,    -  <^  is 

less  than  a,  and  positive.  Now  we  started  by  assuming 
that  a  position  of  I  would  be  found  to  the  right  of  P,  and 
the  positive  result  verifies  our  assumption. 

3.  The  other  position  of  the  point  of  equal  illumination 
is  either  to  the  right  of  Q  or  to  the  left  of  P  according 


{,n  ADKATic  i:<,)rA'ri(>Ns.  361 

as  w  is  greattM-  or  less  tlian  n.     For, is  positive  ami 

m  —  n 

(jreater  than  a  when  in  —  n  is  positive;  i.e.,  when    nt  >  n. 
Ill   tliis  case  the  second   point  of  equal  illumination  lies 

ma 
beyond  Q.     And  is   negative  when   n  >  7w,   and    the 

second  point  of  equal  illumination  lies  to  the  left  of  P. 

Moreover,  all  this  is  in  accord  with  the  conditions  of  the 
problem ;  for  the  two  lights  will  illuminate  equally  a 
point  between  them  situated  close  enough  to  the  weaker 
light  to  make  up  for  the  stronger  illumination  of  the  other. 
Besides  this  point  there  will  be  another,  situated  on  the 
side  next  the  weaker  light,  so  that  here  again  the  closer 
proximity  of  the  one  light  is  offset  by  the  greater  intensity 
of  the  other. 

4.  Certain  special  cases  are  worthy  of  attention  for  the 
analytical  results  they  give. 

(1.   Suppose  m  =  0;  then  x  =  +  0  or  —  0.  (358,  1) 

Here  while  the  light  at  P  is  infiiiitesimally  weak,  the  two  points  of 
equal  illumination  are  infinitely  close  to  it  on  either  side,  making  up 
over  the  other  light  in  closeness  what  is  lost  in  intensity. 

This  can  be  made  clearer  by  an  example.  Suppose  at  the  distance 
of  one  foot  Q  is  very  bright  and  P  is  tery  weak.  Tlien  at  J  ft.  P  is 
4  times  brighter;  at  \  ft.,  10  times  brighter;  at  j^  ft.,  10000  times 
brighter;  and  so  on.  Thus  by  going  closer  the  illuminating  power 
of  the  weak  light  may  be  made  to  increase  indefinitely.  Of  course 
this  reasoning  assumes  the  source  of  the  light  to  be  a  mathematical 
point. 

(2.   Suppose  m  =  n,  then  ar  =  J  a,  aiul  x  .  (358,  3) 

Here  one  position  of  I  is  midway;  the  outside  position  is  at  infin- 
ity, for  it  is  only  at  such  a  distance  that  the  advantage  one  light  has 
over  its  equal  is  reduced  to  an  infinitesimal. 

(3.   Suppose  a  =  0,  then  ])<)tli  values  of  ac  =  0. 

This  is  the  same  problem  as  the  general  case,  only  in  miniature. 
The  value  of  a  and  the  two  values  of  x  are  by  no  means  equal. 
What  is  conveyed  by  representing  all  three  by  the  limit  0,  is  that  all 


362  TEXT-BOOK    or    ATXJKP.IIA. 

are  infinitesimals,  and  that  the  two  points  of  equal  illumination,  one 
between  and  the  other  outside  of  the  infinitesimal  distance  between 
P  and  Q,  are  infinitely  close  to  them. 

(4.   Suppose  a  =  0,  and  ni  ^^  »,  then  x  =  0,  and  ^  a.         (385,  5) 

Here  the  lights  are  infinitely  close  to  each  other,  and  the  one  point 
of  equal  illumination  is  midway,  and  of  course,  infinitely  close  to  P 
and  Q,  while  the  other  position  is  indeterminate,  since  we  do  not 
know  the  laws  by  which  the  quantities  approach  the  limits.  If  the 
lights  are  practically  equal,  and  at  practically  the  same  point,  they 
will  illuminate  any  and  every  point  on  either  side  alike. 

360.  Validity  of  Processes  of  Solution.  There  are  certain 
]»i'0('es.ses  in  the  solution  of  equations  which  deserve  special 
consideration  lest  the  student  be  led  into  serious  error  in 
using  them. 

1.  I?i  ihc  erf I'iicf hill  (if  fill'  xdiiii'  root  of  fliP  tiro  memhers 
of  'III  equation. 

(1.    Thus,   while   (5  —  2)'^  =  (±  :])-,  5  —  2  =  4-3  only  and  not 
— 3;  that  is,  the  roots  of  both  members  must  be  taken  with  the  same 

sign. 

(2.    Fiom  ./■■-  —  2  us  +(/-  =  0 
./•  -  r/  =  ±  0 

and  there  are  iiKo  values  of  ./•  each  equal  to  a. 


(3.    The  Equation  12  .r  —  .5  +  2  V  30  x-  —  27  .r  +  5  +  3  a*  —  1  = 
-  2.  (See  Ex.  3,  327) 


gives  V  12  .<■  -  .-)  +  v^  3  .r  -  1  =  V  27  .r  -  2.  (Ax.  (5) 

As  shown  in  327,  while  x  =  3^  satisfies  the  first  equation,  it  does 
not  satisfy  the  second,  except  when  the  signs  of  the  radicals  are 
taken  in  a  certain  way.  This  anomaly  is  caused  by  the  presence  of 
imaginaries. 

2.  An  equation  ohfaiiwd  hy  squavinr/  or  cnhing,  etc..  tlw 
tiro  Diemheri^  of  o n  ("/luifioii  irlll,  in  f/e)i.eraf  hare  root.^  ir/iic/i 
(to  im^  s(iti><fi/  tJie  oi'if/iniil  cjiKitKiii . 

(1.  To  prove  this  for  scpiariug  take  X  and  X'  for  the 
two  members  of  the  giv^en  equation,  either  or  both  of  them 
functions  of  the  unknow^n,  but  neitlier  zero. 


QL'AI)l;All;'     l.<M    A  IK  >N>.  8«)8 


X   =X' 

(Hyp.) 

x-^  =  X'^ 

(Ax.  5) 

V2_X'^  =  0 

(Ax.  2) 

V -■_>;■-•  _(X+X')(X - 

-\')=0. 

r*\  the  ilieoiy  ot  (•quations,  to  tiiul  tlie  roots  of  this  equa- 
tion the  two  fiictors  are  in  turn  set  e(iual  to  zero.  (353) 

.-.  X  —  X'  =  0,  and  X  -f  X'  =  o. 

Now  the  hrst  of  these  is  the  orhjindl  equation,  while  tlie 
second  is  a  new  equation  with  new  roots. 

Thus,  Ex.  0.  329. 

\  4  or  -H  5  —  ViC  =  Vcc  •+-  8 
Squaring  and  transposing,  and  afterwards  factoring 


4  jt  +  5  -  2  VlWVhx  ^  X  -  (x  +  3)  --  ( V4  J-  -I-  5  -  \/.r  + 
\  s  +~3)  (Via* +  5  -  V?  —  y/x  +  A)  =  0. 

the  second  factor  bein.!?  the  oriijinal  equation,   uliilc  tlu'  first  fur- 
nishes new  roots. 

(2.    To  show  tlu'  sain(^  for  cultni-  ;iii  t'(pi;ii  ion, 

X  =  X'  (Myp.) 

X-'  =  X''^  (Ax.  T)) 

X3  _  X'«  =  (X  -  X')  (X-  +  X>^'  +  >^"',^  =  <>•  ^134,  L>) 

Here,  besides  the  old  equation,  tluM-e  is  the  factor  X-  -|- 
XX'  +  X'-  =  i)  whi(;h  lias  new  roots  of  its  own.  And  so 
for  other  jKJwers. 

To  ilhistrate  the  principle  further,  let  us  solve  the  (M| nation, 

v^TsT+lo  =  i  (x  +  8 )  (  V^  -  1). 
x«  +  5  X  +  10  =  x^  +  9x2  +  27 X  +  27  (Ax.  :,) 

9  x*  +  22  X  =  -  8 
9.x2-f-22x+   (V)'^  =  V 
3x+  V  =  ±i 

X  =  -  2,  or  -  ^. 

Upon  substituting  these  values  in  the  original  equation,  wo  readily 
find  that  neither  of  them  satisfies  it.     They  do  satisfy,  however,  the 


S64  TEXT-BOOIv    OF   ALGEBRA. 


equation  framed  as  X^  +  XX'  +  X'^  =  0,  viz.,  V{x^  +  5  a;  +  19)2  + 
i  </x^  +  ox  +  19  {x  +  3)  (V^=^3  -  1)  +  i  (a-  +  8)  2(  V^  -  1)2  =  0. 

Tims,  \/-8-lQ+l9=l;x  +  3  =  1;  or,  1  +  i  x  1  x  1  x  (V-^ 
-  1)  +  i  (- 2  -  2  V- 3)  ^  0. 

So  also  for  the  value  —  |. 

3.  An  equatio7i  may  not  he  multijylied  or  divided  through 
by  a  function  of  the  unk^iown  which  equals  zero,  or  infinity, 
when  the  value  of  x  is  substituted  i7i  it. 

(1.  In  the  equation  ax^  +  2  bx'^  —  ex  =  0,  a?  is  a  factor 
which  shows  that  one  of  the  roots  of  the  equation  is  zero 
(353).  To  proceed  with  the  sokition,  x  is  divided  out,  but 
zero  must  appear  along  with  the  others  as  a  root  of  the 
equation. 

(2.  In  like  manner  it  is  not  allowable  to  divide  out  by 
one  or  more  factors  of  the  form  (x  —  a),  and  take  no  further 
account  of  the  roots  they  give. 

Thus,  in  3  (ic  —  a^  =  5x  (a  —  x) 
X  =  a  is  one  of  the  roots  of  the  equation. 

(3.  It  is  not  allowable  to  multiply  an  equation  through 
by  x,  thus  introducing  the  root,  zero,  when  this  root  does 
not  by  right  belong  to  the  equation. 

Thus,  x-3  _  ^  _  2  _  8-6x  +  x^ 

.5  x2        X  x^ 

x^  -  30^2  -  40  =  10  ic  -  40  -}-  30  x  -  5  x'^  (Ax.  3) 

cc3  +  2ic2_40x  =  0. 
Since  the  last  equation  contains  x,  zero  is  one  of  its  roots.     But, 
upon  simplifying  the  given  equation  by  transposing  —  -72,  cancelling, 
and  afterwards  multiplying  through  by  x,  we  have 
x2  -  3  X  =  10  4-  30  -  5  X 
or,  x2  +  2x  —  40  =  0 

and  this  equation  does  not  contain  x  as  a  factor  at  all.     Putting  x 

8        3 
=  0  in  the  original  equation,  after  cancelling  —    ,,  — -  is  the   left 

x2       5 

Q 

member,  and  -  +  1  =  x,  the  right  member;  i.e.,  an  infinite  quantity 

equal  to  a  finite  quantity,  which  is  absurd.  Hence  x  =  0  does  not 
belong  to  the  original  equation. 


QUADRATIC    ICQUATIONS.  365 

(4.  It  is  allowable  in  clearing  of  fractions  to  multii)ly 
tliruugh  by  functions  of  x,  provided  that  in  so  doing  no 
alien  values  of  the  unknown  are  introduced.  To  niultii>ly 
or  divide  by  any  known  finite  quantity  is  therefore  always 
adniissil)le. 

i)X  —  a  X      _4x  —  2a4x  —  n 

TTV  ■•"  xTJ)  ~  ~~x  +  i(    ■*■  "j-TlT  "^  ^• 

Clearing  and  simplifying,  we  have 

(x  +  a)x  =  ix  +  a){4x-  a).  (Ax.  ;J) 

(x+ «)x-(x  +  «)  (4x-a)  =  0.  (Ax.  2) 

{x-\-  a)(x  —  (4x-a^)  =  (x  +  a)  (—Sx-\-a)  =  0.  (136,3) 

Hence  x  +  a  =  0,  or  x  =  —  «  is  a  root  of  the  equation.  But  the 
first,  third,  and  fifth  teniis  of  the  given  equation  taken  together 
vanish.  Consequently,  it  was  not  necessary  to  nuiltiply  through  by 
X  +  «,  and  when  it  was  done,  the  extraneous  root  x  =  —  a  was 
wrongfully  inserted. 

y +-      1 +i 

2/  V 

(.').    Clearing  the  equation   r  H j  =  2  of  fractions 


we  get 


•^     y 


1  1  11  11 

clearing  again,  collecting  and  arranging 

4  y2  +  4  y  =  8 
.'.  y  =  1  or  —  2. 

But  while  y  =  —  2  satisfies  the  original  equation,  y  =  I  does  not. 
To  find  the  reason  for  this,  we  simplify  the  left  member  (See  Arts. 
183,  162),  obtaining 

y--\  y-\ 

cancelling,  we  have  left, 

2  2  1/2  \ 

The  second  factor  — j_-  j   +2  =  0  gives  y  =  —  2  as  above.     The 
first  factor  -^Z\  equals  «  when  y  =  I.  (368, 3).    Thus  in  clearing  of 


366  TEXT-BOOK   OF    ALGEBRA. 

fractions  the  equation  was  multiplied  through  by  a  factor  (y  —  1)2, 
which  for  a  particular  value  of  y  equals  x .  Hence  we  have  no 
right  to  expect  that  this  value  of  y  will  satisfy  the  given  equation. 

Tlie  principle  stated  at  the  beginning  of  3  of  the  present 
article  is  of  general  application  and  explains  the  apparent 
difficulty  in  2,  as  well.  It  makes  clear  also  the  fallacy  in 
the  pretended  proof  of  the  arithmetical  absurdity  that 
1=2.     The  "  proof  "  is  as  follows  :  — 

If        a  =  X,  then  ax  —  xr  =  cfi  —  x^ 

X  (a  —  x)  ^  (a  +  x)  {a  —  x) 

x  =  a  +  X  m    (Dividing  out  by  {a  —  x)) 

x  =  x  +  x  =  2x  (Since  a  =  x) 

1=2  (Dividing  out  by  x). 

Here  the  second  equation  is  divided  through,  according  to  the 
hypothesis  by  zero,  and  it  does  not  follow  that  the  next  equation  is 
true  for  the  same  value  of  x.  Thus,  0  X  (any  number)  =  0  X  (any 
other  number). 

It  may  be  said  in  closing  this  article  that,  if  a  student  is 
at  any  time  unable  to  tell  a  2>^"^ori  whether  a  certain  value 
of  the  unknown  belongs  to  the  initial  form  of  the  equation, 
he  has  this  recourse,  that  he  can  substitute  in  that  equation 
and  find  out. 

SECTION  III, 

Equations  of  Higher  Degrees  Solved  like  Quadratics. 

361.  Equations  containing  but  one  unknown  solved  like 
incomplete  quadratic  equations.  —  Binomial  Equations.  — 
Incomplete  quadratic  equations  and  similar  equations  of 
other  degrees  are  often  called  Binomial  Equations,  because 
when  reduced  they  consist  of  but  two  terms. 

1.    Solve    2  .t4  _  9  =  7  -|_  a;4^ 

x'^  =  16  (Ax.  a) 

:r2  =  JU  4  (Ax.  6) 

X  =  \/T"4  =  i  ^ ;    or  V^=^  =  -t  2  V-1 . 

(Ax.  (;.  353.  <i) 


QUADRATIC   EQUATIONS.  307 

2.  Given    x^  —  a'\ 

111  this  example  one  sees  p)nma  facie  that  x  =  a  is  one 
root.  Transposing  the  «^  to  the  left  member  and  dividin- 
tlirough  by  X  —  a  (353.  ' ).  u  •'  Iiavi 

x^  _  a»  =  (x  —  a)  {X-  +  ax-^  a^)  =  0. 

Setting  the  second  fa(5tor,  x:^  -\-ax-\-  a*  equal  to  zero  (^353), 

4  x'  -h  4  (tx  =  -  4  «■- 

^  X'  -\-'Xax  -\-  a-  =  —^a^ 

'>:x^a  =  ia  V-3 

These  two  roots  along  with  <(  form  the  three  roots  of  the 
equation  (353,  a).     These  values. 

—  a  -{•  a  V—  .')     —  (I  —  (t  V  —  :; 
«,  2  '  2  ' 

are  sometimes  called  the  three  cube  roots  of  the  quantity  a^ 

or,  if  "  —  1.  "f  iniity. 

3.  ./• "  =  "  ;  iHMpiired  to  tind  x. 

x»*  =  a"  (Ax.  5) 

n 

x  s=sa  m.  (Ax.  6) 

NoTK.  —  This  is  the  general  ease.  If  the  exponent  of  the  un- 
ktiov.ii  is  negative,  make  it  positive  by  transference  to  the  opi)osit(' 
tt nil  of  its  fraction. 

4.   Given px~^  =  q- 

SOLUTKIN  '\   =  U  \     p^nx^'.    x^  =  ^  ',    X  =  ^ .       A/fS. 

6.     X^=  1. 

SrG(iEsTiox.     (.r«  —  1)  =  (./••■*  -\-  1)  (.!•*  —  1)  =  0. 

6.    X*  =  —  a*. 

Suggestion. —Transjwse  —  n\  and  then  factor  .r*  +  «*  (134,  5). 
The  factors  set  equal  to  a^ro  will  give  the  four  roots. 


368  TEXT-BOOK    OF    ALGEBKA. 

7.  Find  only  the  real  roots  of  this  and  the  following  • 

9        x'' 

8.  4.*-' =  500.  9.   ci-^==81. 
10.    ^x  ^  =  '2  V2. 

.T-  —  na       nx^  —  b 


11     '^   -r  '^-  -r  o    I    -^t^  1-  ♦^  —  o  ^^  2 


12. 
13. 


a;'-  —  a         x'^  —  b 
a  b 


x^^  —  b       x^  —  a' 

14.  a;S  — 32=0. 

15.  x^  =  390625. 

16.  (ic2  +  3  ^  +  2)2  =  (cc2  „  5  ^  .|..  10)2. 

17.  (ic2  _  3  ^  _  10)2  _  15  ^^  _  5^2^ 

362.  Equations  containing  but  One  Unknown  Solved  like 
Complete  Quadratic  Equations. 

There  are  two  type  forms  of  these  equations : 
ax^^  -\-  bx"  =  G,  the  simple  form, 
[f(^)Y"-\-b[f(^)T  =  ^5  tl»e  complex  form, 
f(x)  in  the  latter  meaning  any  f  miction  of  x. 
1.    Solve  3  ic«  +  42  :r«  =  3321. 

x''  ^\^x}  ■=  1107.  (Ax.  4) 

a;6  _f_  14  a;3  _|_  49  _  1155  ^335^ 

£C3  _|_  7  =3    _|_  34 

x^  =  27  or  —  41 

X  =  a/27  or  v'il. 

If  in  the  three  answers  of  Ex.  2  of  the  last  article  we 
put  a  =  V27  =  3,  or,  V—  41,  we  get  the  six  roots  of  the 
given  equation,  it  being  of  the  sixth  degree. 


2. 


QUADUATIC    i.C^U Allocs 6. 

87/ '^4-1 


_  14  y2  =  _  1 


369 


(Ax3.  3  and  2) 


—  14  y2  +  49  =  48 

-7=  ijN/48 
=  7  i  V48  _ 
//  =  i  (-*  i  V3). 


(323) 


3.   Given  (3x'-\'Sx^-  20)^  _  9  (3  a;*  -f  8  a;^)  -  2880  =  0. 

(Ct.  345,  4  and  Ex.  16  in  346.) 

(3  a;*  +  8  y'  -  1^0)-  -  1)  (3  r*  -f  8  .r2  -  20)  =  3060. 

To  save  writing,  put  z  for  3  x*  +  8  ic'^  —  20  :  then, 


«2  _  9  ;g  =  3060 

4«2-.,%;^  =  12240 

4  «2  _  (    )  4-  81  =  12321 

2;s:  =  9illl 

«  =  60  or  —  51 

.-.  3a;  *  4-  8  x'^  -  20  =  60 

9  a-* +  24  a;-'' +  16  =  256 

3a!=^-h4  =  ±16 

aj^*  =  4  or  —  Y  

a;  =  i  2  or  f  V^^IF. 


(334) 


(334,  335) 


The  other  four  values  of  x  may  be  found  by  using  the 
second  vabie  of  z,  i.  e.,  from 


3  x*  +  8  a;2  —  20  ==  -  51. 

4.   5a-4-|-8a-J  =  13 

25  a:4  +  40  a-l  +  16  =  81 
6x14-4  =  4-9 

13 
5 
28561 


(334,  335) 


xl  =  1  or  ~ 
a;   =  1  oi 


62r) 


370  TEXT-BOOK    OF   ALGEBHA. 

5.  ^G  _  7  ^3  ^  8^  7^   x^  -74:x''=  -  1225. 

6.  z"" -5z^-\-4:=0.  8.    17?/* -1=16/. 

In  this  and  some  of  the  following  only  the  real  roots  are 
given  in  the  answer. 

9.    3  7/4  -  7  ?/2  =  25. 

10.   x^  -  24  V^=  81,  ' 


11.  a;2-3  =  6  + V:c--3. 

12.  5  ic  =  39  H-  2  Vx. 

13.  aic2»  -f-  te"  =  c. 

14.  V^  +  4  a/^  -=  21. 

15.  -y/x  +  l  -\/^=3oi). 

16.  12aj-i-a;-i  -2-*  =  0. 

17.  ic  +  Vx^  =  6  Vic. 

18.  ^  V8  =  2  ic  Vl^^'. 


19.    Vic2  _  8  i»  +  31  +  (x  -  4)2  =  5. 
Suggestion.  —  Solve  like  Ex.  3. 


20.    2  Vx^  -  3ic  +  11  =  ic2  _  3^  ^  8 

21.  L  +  8y^^^42-| 


22.    2  ic2  _^  3  ic  -  5  V2  x'^  +  3  cc  +  9  +  3  =  0. 
112 


23-    (2^-4)2-8  +  (2:z;-4)*' 
Suggestion.  —  Put  z  ^  {2x  —  4)2. 

1        17 

24.    Given  0)2  _|_  __  _. 

Suggestion.  —  Add  2  to  both  members  and  apply  Ax.  6 

Note.  —  This  equation  is  one  of  a  class  called  recijjrocal  equa- 
tions, whose  roots  are  reciprocals  of  each  other.  Thus  in  this 
example  the  roots  are  2  and  ^,  and  —  2  and  —  |. 


QUADRATIC  EQUATIONS.  371 

25.  l*rove  by  substitution  that  -,  —  r.aud  —     are  roots  of 

T  )' 

the  equation  w'  -}-     ..  =  a  if  r  is  a  root,  i.e.,  if  /•-  -| ^  =  «. 

26.  k;  //■=  4-  \,  =  i^«. 

27.  (u;  -  a)  (./•  _  ^)  (a;  —  c)  -|-  f?/>r;  =  0. 

28.  (a-j-b-\-jr)  (a-\-b-x)  (a-O-^x)  {a-'b-x)=^  i). 

After  solving  this  juoblem  by  nmltii)lying  it  out,  solve 
it  by  352. 


29.  x*  +  5a-2  +  4  Vic*-f5a:^=60. 

30.  a:»-6x-9  =  0. 

SLOGE.STION.  —  Put  y  +  z  —  j[,  then  (y  +  zf  —  6  {y  +  z)  =  9. 
Developing  this  we  have  y*  +  2^  +  (3  yz  —  6)  (y  +  2)  =  9.  Now,  since 
the  original  equation  contains  but  one  unknown,  and  we  have  just 
introduced  <iro,  we  are  at  liberty  to  make  another  assumption,  that 
is  form  another  equation,  as  two  equations  are  necessary  when  there 
are  two  unknowns.  Let  us  then  suppose  further  that  '6  yz  —  (i  =  0. 
Whereupon  the  first  equation  reduces  to 


(Hypothesis  above; 


(1) 

y3  +  28  =  0 

(2) 

:iyz-{)  =  0 

(-^) 

2 

z  =  - 

y 

(1) 

/2\  3 

*'  +  (,7)  = » 

yo  4-  8  =  9  2r* 

0  .  «1       81              49 

9            7 
2^~2=±2 

2/«  =  8  or  1 

y  =  2  or  1 

(2.) 

2  =  1  or  2 

x  =  y  +  2  =  3.     Ana. 

This  is  called  Cardan's  solution  of  a  cubic  equation.     The  other 
two  roots  may  be  found  from  y'  =  8  as  in  Ex.  2,  361. 


872  TEXT-BOOK   OF    ALGEBRA. 

31.    x^  —  2x^-2x^-\-3x-10S  =  0. 

SuGGKSTiON.  —  Sometimes  equations  of  this  kind  can  be  arranged 
in  the  quadratic  form  by  extracting  the  square  root. 

re*  -  2  .^3  -  2  x2  +  3  X  -  108  (x'^  -  x 


2  x2  -  X 


-  2  x3  _  2  x'2 

-  2  x3  -f-     a;2 


-  3  x^  4-  3  X  -  108. 


Hence  x^  -  2  x^  -  2  x^  +  S  x  -  108  ^  {x^  -  x)'^  -  (3  x2  -  x)  -  108 
=  0. 

To  proceed  with  the  solution  put  z  ^  x-  —  x. 

32.  x'  -6x^  -{-6x^  -\-12x  =  60. 

33.  4  cc*  -f-  ^  =  4  x^  -h  1. 

X  •{■  yjx        X^  —  X 

34.     —r  =  -j • 

a?  —  yjx  4 

Suggestion.  —  Divide  through  by  x  +  \fx.  The  roots  are  then 
found  to  be,  1,  4  and  |(—  6  ±  2  V—  7),  notwithstanding  that  in  this 
instance,  the  first  root,  Vx  =  —  1,  makes  x  +  Vx  zero.     (360,  3.) 

35.  a^*  —  4  :zj3  +  6  ^2  —  4  cc  -I-  1  =  16. 
Suggestion.  —  Extract  the  fourth  root. 

8  a* 

36.  x{x  —  2  a)  =  -2 7. —  +  7  a\ 

^  '       x^  —  2  ax    ^ 


\    X    —   4: 


37.    "    •    "       "     '-  ^-  ^ 


4  Va^  -  48 

38.  X  =    :r^ — 

X  —  18 
Suggestion.  —  Clear,  add  4  x  +  49  to  both  sides,  and  apply  Ax.  0. 

39.  (3  -  xy  +  (4  -  xy  =  17. 

Suggestion.  —  Put  3  —  x  =  ?/,  and  4  —  x^v.    Then  (1)  ^i*  +  v* 
=^17.     (2)  u  —  V  =  -^  1.     Raising  both  rnenibers  of  (2)  to  tlie  fourth 


QUADRATIC   EQUATIONS.  873 

power,  and  subtracting  from  (1),  we  have  4  uh  —  0  mV-  -\-  4  uv^:=. 
4ur  ( i(-  —  2  ur  -h  r-)  +  2  mV  =  10. 

4  uv  (1)  -f  2  uh'^  =  16 
mV  +  2  Mc  +  1  =  9  <334,  335) 

«D  =  2  or  —  4. 

Having  the  values  of  u  —  v  and  wc  the  solution  Is  continued  as  in 
342;  whence  i/  =  1,  r  =  2,  which  give  jf  =  2;  and  m  =  —  2,  c  =  —  1, 
which  give  x  =  5.  The  otlier  value  of  uv  leads  to  imaginary  values 
of  J-. 

40.    (y_4)*-h(l4-y/  =  97. 

General  Remark.  —  Examples  of  higher  equations 
solved  like  quadratics  occur  in  the  most  diverse  forms  and 
are  solved  by  the  greatest  variety  of  methods.  One  or 
other  of  the  suggestions  of  tliis  article  will  often  be  found 
helpful. 

363.  Simultaneous  Equations  of  Higher  Degrees  that  may 
be  Solved  like  Quadratics. 

a.  Wlien  two  equations  are  combined  the  number  of  sets  of  roots 
ouglit'  to  o(iual  the  product  of  the  numbers  denoting  their  degrees. 
Tims  combining  a  cubic  with  a  quadratic  should  give  six  sets  of 
roots.  For,  solving  the  cubic  for  one  of  the  unknowns,  there  would 
result  three  roots,  each  of  which  substituted  in  the  quadratic  would 
make  three  equations  for  the  other  unknown,  giving  in  all  six  values 
for  it.  However,  such  problems  as  can  be  solved  are  degenerate 
totiii^,  and  very  often  give  a  less  number  of  answers. 

b.  These  problems  are  so  varied  in  character  that  no  satisfactory 
classification  can  he  given.  Notwithstanding  the  variety  of  forms 
of  the  problems,  certain  methods  of  solution  are  worthy  of  distinct 
exemplification.  Sucli  are:  Substitution;  division;  introduction  of 
new  variables;  comparison;  solving  first  for  functions  of  the  un- 
knowns; extraction  of  roots;  special  methods  for  symmetric  equa- 
tions.    (230,  a.) 

1.  Given  (1)  a;«  -}-  y«  =  133,  (2)  a-y  =  10.  (228) 

2.  (1)./-*  -h  //"  =  89  (2)  x^  -f  if  =  13.  (342) 


374  TEXT-BOOK    OF    AL(Ji:iiHA. 


^'     (  x^  +  ,f  =  25  xY.         ^'    oF^y-  =  -  '^  ^^^^^-'^   =  ^^• 

4,    U''  +  i/  =  ^2  ^     {x'Jry'  =  m 

\  xy  =3.  '     {x  -\-  y=  11. 

Suggestion.  —  Divide  one  equation  by  the  other. 

7      Jx«  +  //^  =  lo2  ^      (^'^-7/ =  2197 

Ix'  -  xy  +  7/'^  -r  19.        •     '\  X  -  y  =13. 
g      U-4-;y4_^,  ^^      (  x'^  +  iT-y  +  y*  =  2128 

(  ^2  _  y2  ^  ^^  •         (  ^2  _|_  ^^   _j_  y2  ^  J(J_ 

Suggestion.  —  Divide  as  in  Ex.  G. 


11. 


12 


13 


\  x^y^  -f  x'-y^  ==  —  4 

(  x'^y  -\-  xy'^  =  2. 

(  .^3  _^  y^  -^  3cc  -h  3?/  =  378 

(  a^3  _^  ^3  _  3  ^.  _  3  _y  _  324. 

^x+y  ^  35 

i  xh  +7/^  =  5. 
Suggestion.  —  Put  ii  ^^  .ra,  and  v  =  yh.     Then,  n^  =  x,  v^  =  y. 
14       ix\^yh  =  ry  (  VV  +  VF=  20  ^ 

16.     -^  •'^^  +  //"  +  ^  Va-M^  =  45 
<  X*  +  y^  =  337. 

Suggestion.  —  Put  2  ==  Vr-  4-  //'-. 
1^      /  •>•//-  +  xy  =  24 
(  ,r_//-  +  .-/•  =  40. 
Suggestion.  — Factor  the  left  members,  and  eliminate  .r  by  com- 
parison. 

18    <  '^■''  (^  -  //)  =  ^^-  '      19     ^  (1)  ^//  +  '-r  =  12 

(  .,.2  (2  X  +  3  //)  =  28.       ■      (  (2).r  +  xy''  =18. 
St^ggestion.  —  Divide  one  equation  by  the  otlier. 

20   ^  ^  ^//  =  ^^^  -  •^'^//'       21     ^  "^^^  +  y  =  21 

(  ,r  +  y  =  0.  ■      i  xY  +  2/'  =  '"33. 

SrfJfiKSTiON.  —  Solve  first  for  xy~  =  .r',  and  ?/. 


QU ADliATlC    EQUATIONS. 


375 


22. 


(  J-'  -f  ,/3  =  ISi). 


<  .,  >  _  ,/i  ^  (J  Va:*  —  /  =  16 

^  -  +  ''7/  +  y^  =  2. 

(x-  +  1)  i^  =  o^y  -  744. 
Suggestion.  —  Eliminate  the  left  members,  and  solve  for  xy. 


23. 
24. 


2^-     t(2)^!/(a^  +  2/)  =  70. 

SuoGESTiox.  —  Add  three  times  (2)  to  (1),  and  extract  the  cube 
root  of  the  members  of  the  new  equation. 

/  u-«  -  7/«  =  56 
26.     3  16 


oci/ 


27. 


+  /r  = 


13 


Xt/  = 


28. 


(a-  +  //  =  3 
<.,.4_^  //  =  17. 

J«»i:ft(;KSTiox.  — These  equations  are  symmetrical  in  x  and  y.  Let 
X  ==  M  +  r,  autl  ?/  =  ff  —  r.  Then  x  +  y  =  2  u  —  :),  and  ?i  =  3. 
Thei-efore  x  —  I  +  r,  and  ?/  =  3  —  r,  and  u-»  +  y*  =  (3  +  r)^  +  (3  —  r)» 
=  17.  Upon  developing  the  terms  of  the  last  equation  it  is  readily 
solved  as  a  quadratic. 

29       f^Z/=«(^H-2/) 

Divide  these  equations  through  by  xy  and  x*y*  respectively,  and 

solve  for  -  and  ^  • 

X         y 

30.  {\)  X  -  y  =2  {a  •-  z)  (2)x^  -  if  =  ii  (a^  -  x'^  (3)  a-' 
_y8=7(««-';i«). 

Suggestion. —  Divide  (2)  by  (1),  giving  (4);  solve  for  x  and  y 
from  (1)  and  (4),  and  substitute  in  (.i)  -f(l). 


376  TEXT-BOOK   OF   ALGEBRA. 

31.    (1)  X  -h  y  =  15  (2)  xij-\-xz-\-  yz=2H  (3)  xyz  =  756. 

756 
Suggestion.  —  P'iiid  z  =  , ..  _    s,  and  ?/  =  15  —  .t,  and  substitute 

these  values  in  (2). 

364.    Problems  Depending  on  the  Solution  of  Equations  of 
Degrees  other  than  the  First  or  Second. 

1.  A  bill  to  contain  100  bu.  of  grain  is  made  3i^  times  as 
long  as  wide,  and  2f  times  as  deep  as  wide.  If  one  cu.  ft. 
contains  |.bu.,  what  ninst  be  the  dimensions. 

2.  Two  cubic  vessels  together  hold  407  cu.  inches.  When 
one  vessel  is  placed  on  the  other  the  total  height  is  11  in. ; 
find  the  contents  of  each. 

3.  The  product  of  the  sum  and  difference  of  two  numbers 
is  8,  and  the  product  of  the  sum  of  their  squares  and  the 
difference  of  their  squares  is  80.     What  are  the  numbers  ? 

4.  One  number  is  8  times  another,  and  the  sum  of  their 
cube  roots  is  12.     What  are  the  numbers  ? 

5.  Pind  two  numbers  whose  product  is  15,  and  the  dif- 
ference of  whose  cubes  is  */  times  the  cube  of  their  differ- 
ence. 

6.  The  sum  of  two  numbers  is  6,  and  the  sum  of  their 
5th  powers  is  1056.     Find  them.     (See  Ex.  28,  last  article.) 

7.  The  product  of  two  numbers  is  10,  and  the  sum  of 
their  cubes  is  133.     Required  the  numbers. 

8.  A  man  draws  a  certain  quantity  of  vinegar  out  of  a 
full  vessel  that  holds  256  gals. ;  and  then,  filling  the  vessel 
with  water,  draws  off  the  same  number  of  gallons  as  before, 
and  so  on  for  four  draughts,  when  there  were  only  81  gals, 
of  pure  vinegar  left.  How  much  vinegar  did  he  draw  at 
each  draught  ? 

9.  B  has  33  more  than  the  square  of  the  number  of  dol- 
lars A  has,  and  the  product  of  the  numbers  expressing  the 
fortunes  of  each  is  1330.     How  many  dollars  has  each  ? 


(,)l  .\i>i;.\'l'ic    KC^lATIO.NS.  377 

10.  A  box  measure  has  a  scjuare  bottom,  and  a  bin  lias  for 
its  respective  dimensions  the  S(]uares  of  the  dimensions  of 
the  measure.  If  now  the  bin  is  full,  and  7  measures  full  be 
taken  from  it,  1710  cu.  in.  will  remain.  Moreover,  the  num- 
ber of  square  inches  in  one  of  the  sides  of  the  measure 
diminished  by  the  number  of  linear  inches  in  one  side  of 
the  bottom  is  12.     Find  the  dimensions  of  the  measure. 

11.  What  two  numbers  are  those  whose  difference,  multi- 
plied by  the  difference  of  their  squares,  is  .32,  and  whose  sum, 
multiplied  by  the  sum  of  their  squares,  is  272? 

SuooESTiON.  — Call  H  llu'ir  sum  and  v  their  diflference. 


FIFTH    GENERAL    SUBJECT. -TOPICS 
RELATED  TO  EQUATIONS. 

(INEQUALITIES,    PROPORTION,     EXPONENTIAL    EQUATIONS,    LOG. 
ARITHMS,    PliOGllESSlONS,  AND    INTEREST.) 


CHAPTER   XXIV. 

INEQUALITIES. 

365.  An  Inequation,  as  the  word  itself  suggests,  consists 
of  two  ineiiibers,  one  of  which  is  greater  or  less  than  the 
other  but  not  equal  to  it. 

We  now  propose  to  study  briefly  the  inequation  in  the 
same  way  the  equation  itself  has  been  investigated. 

a.  The  signs  >  and  <  indicate  that  the  quantities  between  which 
they  are  placed  are  unequal,  the  opening  of  the  character  being 
toward  the  greater.  By  (jreater  is  meant  higher  up  in  the  scale  in 
the  positive  direction. 

.  Thus  6>4,  3>-2,  -  11  >  —  15,  etc.  Hence  if  a>h,  then 
a  —  h  is  always  positive.  In  the  examples  just  given,  6  —  4  =  2, 
8  -  (-  2)  =  5,  -  11  -  (-  15)  =  4,  etc. 

366.  Properties  of  Inequations.  The  inequation  has  sev- 
eral of  the  properties  of  the  equation,  hut  not  all. 

1.  If  the  same  number  be  added  to  or  subtracted  from 
both  members  of  an  inetpiation,  the  new  inequality  will  be 
like  the  old ;  i.e.,  will  retain  the  same  sign. 


Thus,  7  >  5;  and  by  adding  4,  11  >  9 ; 

2  >  _  IK),  and  by  adding  -  12,  -  10  >  -  42  : 

—  ^2  >  —  13,  and  by  adding  17,  5  >  4; 

-  (•)  <  0,  and  by  adding  —  18,  —  24  <  —  18. 
and  so  on. 

It  is  evident  from  these  examples  that  both  members  will 
be  clianged  aiiL-f,  or  merely  shoved  along  in  the  scale  whether 
by  adding  or  subtracting,  and  the  new  inequality  will  be  of 
the  same  kind  as  the  old.  A  general  demonstration  of  the 
theorem  is  as  follows  : 

a  >  h  (Hyp.) 

a  —  h  is  ])()sitive.  (a  of  last  Article.) 

w  i  c  —  (^  -I-  c)  is  positive.  (the  c's  cancel.) 
.:a  ^c>  b^c.     Q.  E.  1). 

Scholium. — We  learn  from  this  that  a  term  can  l)e 
transposed  from  one  side  of  an  inequation  to  the  other  by 
(^hanging  its  sign  (210). 

2.  An  inequation  multi})lied  or  divided  through  by  the 
same  positive  multipliei*  or  divisor  will  stand  as  it  did  be- 
fore; whereas  if  the  multiplier  or  divisor  be  negative  the 
new  inequation  will  subsist  in  the  routmnj  sense ;  i.e.,  if  the 
left  meml)er  were  greater  before,  it  will  be  less  now,  and 
conversely. 

A  little  consideration  will  show  that  this  theorem  is  true 
for  the  positive  multiplier  or  divisor,  and  then  also  for  the 
negative,  since  the  greater  a  negative  number  is  numeri- 
cally, the  less  it  is  algebraically.  As  a  special  case  of  this 
j)rinciple,  if  the  signs  of  the  terms  of  an  inequation  be 
changed  throughout,  the  sign  of  the  inequation  nnist  be 
changed. 

3.  If  the  two  members  of  an  inequation  are  both  j)ositive 
numl)ers,  when  they  are  raised  to  any  integral  power  the 
inequation  will  still  hohl  as  before.  If  the  two  members  of 
an  inequation  are  both  negative  numbers  (or  at  least  the 


380  TEXT-BOOK    OF    ALGEBRA. 

less  number  is  negative),  when  they  are  raised  to  an  odd 
power  the  inequation  will  still  hold  as  before.  But  if  the 
two  members  of  an  inequation  are  both  negative  numbers, 
Avhen  they  are  raised  to  an  even  power  the  new  inequation 
will  subsist  in  the  contrary  sense.  Let  the  student  test 
these  cases  by  various  examples. 

Eaising  the  members  of  an  inequation  the  lesser  of  which 
is  negative  to  an  even  power  gives  rise  to  an  amhiguity. 

Thus,  5  >  —  2,  upon  squaring  gives  2^  >  4, 
Avhile,  5  >  —  8,  upon  squaring  gives  25  ^  64. 

Evidently  in  this  last  case  the  greater  member  must  also 
be  the  greater  numerically. 

4.  If  any  root  of  the  two  members  of  an  inequality  both 
of  which  are  positive  is  extracted,  or  any  odd  root  where 
both  are  negative,  the  resulting  inequality  will  be  like  the 
old,  providing  in  the  former  case  the  root  of  the  greater  be 
taken  as  positive. 

Test  this  witli  examples. 

5.  Two  or  more  like  inequalities  may  be  added,  just  as 
equations  are,  and  the  resulting  inequality  will  subsist  in 
tlie  same  sense. 

6.  But  if  one  inequality  l)c  taken  from  another  like  in- 
equality, the  result  is  am'oiguous. 

Thus,  12  >  5  12  >  5 

2  >  1  11  >  1 


10  >  4  1  ;2^  4 

7.  Like  inequalities  whose  members  are  all  ])ositive  may 
be  multiplied  togetlier,  and  tlie  new  inequality  Avill  subsist 
in  tlie  same  sense. 

To  the  theorems  given,  still  others  niiglit  be  added.  Tlie 
foregoing  are  the  most  important,  and  sufficiently  illustrate 
the  elementary  treatment. 


TOPICH    iiKl.ArKD    TO    EQUATIONS.  381 

367.    Special  Theorems  in  Inequations. 
1.    To  prove  that  the  sum  of  tlie  squares  of  any  two  real 
and  unequal  nuniljers  is  greater  than  twice  their  i)roduct. 
Let  a  and  b  be  the  numbers. 
Now  a  square  number  is  always  positive.     'V\wn 

(a  -by>o 

(except  when  a  =  b,  when  (a  —  b)'^  =  0) 


a^  —  2ab-[-b''>  U 

(114.  1>) 

a^  -\-b^>  2  (lb          (J.  J 

V.  />. 

(366,  1) 

Remark.  —  This  theorem  is  much  used  in  the  demonstratioii  of 
other  special  ones. 

2.  Prove   that   any  positive  fraction  whose  terms  are 
unequal  plus  its  reciprocal  is  greater  than  2. 

Thus,  to  prove  -  +  --  >  2. 

3.  Show  that  a*  -j-  ^*  >  a%  +  ab^,  when  a  -{-b  is  positive. 
SuooESTiox.  —  Divide  the  inequation  through  by  a  +  h. 

4.  l*rove  that  a*b  +  «//  >  2  a%%  when  a  >  b  >  0. 

5.  Prove   that   x^ -{- 1  ^  jc' -{-x,  when  O! -f  ^   i^   positivi; 
and  X  :^  \. 

6.  Show  that  x'^  -\-  y-  -|-  ^^^  >  xy  -\-  xz  -\-  yx,  if  .'•.  y.  and 
z  are  unequal. 

7.  Prove  that  —  <  —^-J—\~  <  -jf  ^     <  L  <         tlie 

n         n  -\-  q  -{■  s       s       n         y       .s- 

denominators  ti,  q  and  s  being  all  positive. 

8.  Which  is  the  greater,  a^*  +  3  ^*  or  2  ^»  (a  +  /;),  the  let- 
ters being  unequal  and  positive  ? 

9.  Which  is  the  greater,  a*  —  Z**  or  4  a'*  (a  —  b),  wlien 
a>  b  >  0. 

SuooESTioN, -r-fftctor  the  left  member  and  tlivide  through  by 


382  TKXT-BUOK    OF    ALGEBRA. 

10.  Griven  that  a,  h,  and  c,  are  positive  and  unequal,  to 
show  that  («  -{-  b)  (b  -\-  c)  (c  +  a)  >  8  ahc. 

Suggestion. —Divide  the  three  factors  of  tlie  left  member  by 
«,  A,  and  c  respectively,  thus  cancelling  the  same  factors  in  the  right 
member.     Then  multiply  out  and  apply  the  theorem  of  Ex.  2. 

368.    Examples  and  Problems  in  the  use  of  Inequations. 

1.    Given  '2  ,r  -f  i  .x  —  4  >  6  to  fnid  the  limit  of  x. 

Sr(;(;i-:sTi()\.  —Solve  as  an  equation  by  clearing  and  transposing, 
v.iience  it  will  appear  that  x  >  4; 

^x      5        11       7x 

3.    Find  the  limit  for  both  x  and  y  in  (  (1)  ^x  +  4  y  >  30. 

^       |(2)3.r4-2y  =  31. 


X  X 

\'t  -\-h  >  -  -\-  X  to  find  the  limit  of  .r, 

4.  Given  •<' 

X       X       a  —  h 

7-  —  -  >  — ;; —  when  a'y  b  y-  (). 

\b        a.  b 

5.  Show  that  ^^  —  8  £c  -f  22  is  never  less  than  6  whatever 
may  be  the  value  of  xx     (See  357.) 

6.  The  double  of  a  certain  number  increased  by  7  is  not 
greater  than  19,  and  its  triple  diminished  by  5  is  not  less 
than  13;  required  the  number. 

7.  A  certain  positive  whole  number  plus  23  is  less  than 
6  times  the  number  minus  12 ;  and  nine  times  the  number 
niiruis  54  is  less  than  twice  the  number  i)lus  9.  What  is 
the  number  ? 

8.  Prove  that  the  arithmetic  mean  of  two  luimbers  is 
always  greater  than  their  geometric  mean.     Or,  as  a  formula 

it  is  to  jjrovethat   ~.>      >  ^ ab. 


TOPICS    UKLA'PKI)   To    KQUATlUNS.  383 


CHAPTKK    XXV. 

liATIo    AND    rUOPORTION. 

369.  The  Ratio  oi  two  quantities  is  the  (jiiotient  of  th(^ 
first  divided  by  tlie  second.  The  first  quantity  is  called  the 
antecedent,  and  the  second  the  consequent. 

Thus,  the  ratio  3  to  6  is  g  or  \  ;  the  ratio  of   a  +  ?«  to 

m  +  >i2   is  ^-^^,  =  (a  +  w)  ^  (ni  +  n') 

^  (a  -\-  }n)  :  ()n  -f-  7i^. 

a.  Ratios  are  usually  denoted  by  a  colon.  Thus,  a  :b  is  read  the 
ratio  of  a  to  h.  The  colon  is  supposed  to  represent  a  division  sign 
with  the  horizontal  line  omitted. 

h.  Some  writers  regard  the  second  quantity  as  the  dividend  and 
the  first  as  the  divisor.  But  this  usage  is  out  of  harmony  with  the 
idea  that  the  colon  represents  a  division  sign,  and  should  be  aban- 
doned. 


370.  Reduction  of  Ratios.  Since;  a  ratio  is  a  fraction  it 
(111  1m  I'duced  to  other  t*  i  ms  w  ithout  altering  the  value  of 
the  f  taction,  i.e.,  the  ratio.   (^159.) 

_,  ,        a       ma        ^        ^  •     i       - 

Thus,  a:b  =  -i=:  —J  =  j^        In  i)articular,  12  :  4  =  <) :  L\ 

lit 

QUEUV.  —  "Will  it  change  a  ratio  if  the  two  terms  be  increased  or 
diminished  by  any  number?  How,  if  they  are  increased  or  dim- 
inished by  like  parts  of  each  ? 

SuaGESTlox.  —  Multiply  both  li-rms  by  f  l  -f  "- J  or  by  [  J  —  "'-\, 


384  TEXT-BOOK   OF   ALGEBRA. 

371.  Ratios  can  be  compared  in  Value  by  reducing  them  to 
a  common  denominator. 

Tlius,  to  compare  the  values  of  10  :  11  and  23  :  2o. 

10  23     250       253    .^.   ^^.       10      23 

11  >  '  =  '  <  25'  275  <  275  (^^^'  ^^  •"•  H  <  25' 

372.  To  Compound  Ratios  is  to  multiply  their  correspond- 
ing terms  together. 

Thus,  compounding  a:b,  c:d,  and  e  :/ we  have  ace :  hdf. 

a.  When  the  terms  of  a  ratio  are  squared  it  is  said  to  be  the  du- 
plicate  ratio  of  the  first  ;  when  cubed,  the  triplicate. 

h.  Examples  of  the  compounding  of  ratios  may  be  seen  in  com- 
pound i)roportion  in  arithmetic. 

373.  A  Proportion  is  the  equality  of  two  ratios. 

Thus  f  =  4  ;  OY,  a:h  =  c\d-,  ov  a:b : :  c:d  using  a  double 
colon  instead  of  the  equality  sign. 

a.  The  colon  form  is  read  "  a  is  to  6  as  c  is  to  d!,"  while  the  frac- 
tional form  f  =  ^  inay  be  read  in  the  same  loay,  or  as  an  equation, 
"  a  divided  by  h  equals  c  divided  by  cZ."  Indeed,  both  forms  may  be 
read  in  either  way. 

?).  The  first  and  fourth  terms  of  a  proportion  (a  and  d)  are 
called  the  extremes,  and  the  second  and  third  terms  (b  and  c),  the 
means. 

374.  Method  of  Treatment  of  Proportion.  —  To  accomplish 
any  desired  reduction  the  proportion  is  first  written  as  an 
equation,  and  then  transformed  by  axiomatic  processes.  As 
regards  the  colon  form  of  writing  proportions  it  need  never 
be  used.  In  truth,  it  simply  duplicates  the  other  notation. 
If  any  advantage  at  all  attaches  to  it,  it  is  due  merely  to 
the  fact  that  the  proportion  is  Avritten  in  a  single  line. 
Nothing  of  importance  would  be  lost  and  simplicity  would 
be  gained  if  the  colon  form  were  entirely  discarded.  In 
the  study  of  proportion  we  seek  to  familiarize  ourselves 
with  the_  different  forms  in  which  the  same  proportion  can 
appear,  and  the  new  proportions  which  can  be  constructed 
out  of  it.  * 


Toi'irs  i:klati:i)  to  kc^katk )NS.  385 

375.    Transformations  of  a  Proportion. —  Thkiujkms. 

1.  In  a  pi'oj/ortion  f he  product  of  the  ejcfn'ines  is  equal  to 
the  prodiif't  of  the  means.  This  is  the  usual  test  for  a 
proportion. 

Given  a  '.h::r:(l  to  prove  ad  =  be. 

ad  =  hr     Q.  E.  1).  (Ax.  3) 

2.  CoxvER.SK  Thkohkm.  —  If  the  product  of  two  numbers 
is  equal  to  the  product  of  two  others^  one  pair  may  be  made 
the  extremes  and  the  other  pair  the  tneans  of  a  proportion. 

For  if  the  equation  be  divided  througli  by  any  pair  of 
opl)Osite  terms,  a  proportion  results. 

Taking  the  equation  ad  =  be,  or  be  =  ad 

(1 )  J  =  -j  i-<?v  (i:b::c:d (by  dividing  through  by  bd) 
(1')-^  =  -  i.e.,c:d::a:b(     -  -  -  ••     j 


(li)  -  =  -  i.e.,  b:a::d:c (by  dividing  through  by 


ac) 


{2')i^t  i.e.,d:c::b:a(      -  ^'  ''  •      ) 

c      a 

'  (3)  _  =  _   i.e.,  a:c::b:d  (by  dividing  through  by  ?) 
c       a 

(3')  ^~'i   \.i^.,b:d::n..^(      -  "  -  ••     ) 

d      c 

c      d 
(4)  -  =  ^  i.e.,  c:a:'.d:b  (by  dividing  through  by  ?) 

(4')'^=^  i.e.,  f/:/y::6':(M      -  "  '*  "     ) 

b       a 

3.  General  Theorem.  —  Since  if  a:b::c:  d,  then  ad  = 
be  (bj/ the  first  theorem)  ;  a7id  furthermore,  since  all  the  forms 
just  given  foUow  from  ad  =  be  by  axiomatic  processes,  we 


386  TEXT-ROOK    OF    AL(iEBRA. 

leurn   that  if  a  :  h  '.  :  c  :  <h  the)i  all  the  above  forms  of  the 
jji'opoi'tloii  are  likewise  true. 

Evidently  the  four  primed  equations  (1'),  (2'),  (3'),  (4') 
are  essentially  the  same  as  the  others,  the  difference  being 
the  mere  interchanging  of  their  members.  In  translating 
these  formulae  into  the  theorem  form,  it  is  customary  to  call 
a  the  first,  b  the  second,  c  the  third,  d  the  fourth  term  of  the 
proportion.  The  formula  (2')  would  be  read  thus  :  If  four 
quantities  are  in  i)r()portion  taken  in  order,  then  the  fourth 
is  to  the  third  as  the  second  is  to  the  first.  Each  formula 
might  in  this  manner  give  rise  to  a  theorem.  Two  of  these 
changes  in  the  original  proportion  have  been  given  specific 
names,  viz.,  (2)  and  (3),  called  respectively  Inversion  and 
Alternation. 

a.  A  proportion  is  said  to  be  taken  l)y  inrey.sioii  when  the  second 
is  to  the  first  as  the  fourth  is  to  the  third. 

b.  A  proportion  is  said  to  be  taken  by  alte)')i(itlo)i  when  the  first 
is  to  the  third  as  the  second  is  to  the  fourth. 

Queries.  (1)  Will  a  proportion  still  hold  if  all  its  terms  be  mul- 
tiplied or  divided  by  the  same  number  ?     (370,  158.) 

(2)  Will  a  proportion  still  hold  if  the  first  two  terms  are  multi- 
plied or  divided  by  one  number  and  the  second  two  by  another  ? 

(3)  How  can  the  signs  of  the  terms  of  a  proportion  be  changed 
without  altering  its  value  ?     Can  any  two  be  changed  ?'    (157.) 

(4.)  Will  a  proportion  still  hold  if  the  antecedents  be  multiplied 
or  divided  by  one  number,  and  the  consequents  by  another  ? 

(5.)  Will  a  proportion  still  hold  if  the  same  number  be  added  to 
or  subtracted  from  each  of  its  terms  ? 

(6. )  Will  a  ratio  represented  by  an  improper  fraction  be  increased, 
diminished,  or  unchanged  by  adding  the  same  quantity  to  both  of  its 
terms  '?     How  will  a  proper  fraction  be  affected  ?     IIow  unity  ? 

376.    New  Proportions  from  a  Given  One. 

1.  The  Compositiox  Theorem.  — If  four  (piantitie.^  are 
in  iiroportionj  the  sum  of  the  frsf  (.nd  second  is  to  the  first  or 
second  as  the  sum  of  the  third  and  fourth  is  to  the  third  or 
fourth. 


TOPICS    liKi.ATKl)    ro    KIJL  ATloNS.  887 

It  "  .  /'     :  '   :  ./.    then    -        T  ,     /  I      /      - 

For  ;'  +  !  =  ''  +1  ^Ax.  1) 

(1)    ^  =  T/     ^'^-      •  ^^^^^^^ 

Of  coiiisc  this  lu'w  |)i()i)()rti()ii  can  now  be  alternated,  in- 
verted, or  transformed  in  any  of  the  ways  described  in  the 
last  article.     Thus, 

('^)    ^J^  =  !'  O^^-'-     See  Eq.  (3'),  375J 

E(iuation  (4)  can  be  derived  directly  from  -  =  -  (2) 

^'-hl='^+l  (Ax.l) 

n  C 

''  +  "=1+.^     Q.E.D.  fl67) 

II  r 

2.  Tmk  Division  Thkohkm.  —  If  four  inmntifiis  <ir<  in 
iu'i>iiui'tii>n,fhe  difference  of  thr  ji rsf  mid  second  is  fo  flir  jirsf 
or  seroiid  as  the  difference  of  the  third  mid  fourth  is  fo  the 
third  or  fourth. 

It  o  :  f) :  :  r  :  d,  then  ,     ,  ,      . 

(  1/  —  If  :  h  :  :  ■   —  ii  :  il 

For  ''  -  1  ='    -  1  (Ax.  2) 

If  d 

■'''='ZL±    Q.E.D.  (167) 

'/  ll 

(7)   'LszJL  =  '-—±     o.k.d.  ,375./-) 


388  TEXT-BOOK    OF    ALGEBRA. 

3.  The  Compositiox  axd  Division  Theorem.  —  If  four 
quantities  are  In  proportion^  the  sum  of  the  first  and  second 
is  to  their  difference  as  the  sum  of  the  third  and  fourth  is  to 
their  difference. 

If  a  :  b  :  :  c :  d,  then  a -\-  b  :  a  —  b  :  :  c  -{-  d  :  c  —  d 

For,  taking  equations  (2)  and  (6)  of  this  article, 

a  -\-  b       a  —  b  /  \      rr^ 

— J_  = .  (Ax.  7) 

c-\-d       c  —  d  ^  ^ 

1±1='-±1     Q.E.D.  (Z75,b) 

a  —  b       c  —  d 

4.  The  Like  Powers  Theorem. — If  four  qua^itities 
are  in  proportion,  like  powers  of  the  terms  are  also  in  pro- 
portion. 

■    li  a\b::c'.d,  then  a»* :  b"' :  :  c"» :  c^'" 

For  1  =  1  (Hyp.) 

%-  =  %-     Q.E.D.  (Ax.  5) 

5.  The  Like  Koots  Theorem.  —  If  four  quantities  are 
in  proportion,  like  roots  of  the  terms  are  also  in  propoi'tion. 
This  is  proved  in  the  same  way  as  4 ;  instead  of  Ax.  5,  use 
Ax.  6. 

377.    Combinations  of  Two  or  More  Proportions. 

1.  In  a  continued  iwoportion  (i.e.,  one  in  ivhich  three  or 
more  ratios  are  all  equal  to  each  other)  the  sum  of  any  ante- 
cedents is  to  the  .sum  of  their  consequents  as  any  one  ante- 
cedent is  to  its  consequent. 

If  a'.b'.\c\d\\e\f\\(j\h\\  etc.,  then  a  -\-  c  -^  e  -\-  (j  -\- 
etc,  '.b  -\-  d  -{-  f  -\-h  -^  etc.  ::  a-.b::  etc. 

To  prove  it  let  us  suppose  r  to  be  the  common  ratio  ;  then 
a  c  e  a  , 

.*.  a  —  br,  G  =  dr,  e  =  fr,  (j  =  hr,  etc.  (Ax.  3) 


TOPICS    RELATED   TO   EQUATIONS.  389 

Adding  these  equations  (or  any  set  of  them)  member  by 
member 

,,  4.r  +  6  +  <7  +  etc.  =  {h  +  d  +/+  h  +  etc.)  r      (Ax.  1) 
g-f  c-f  g  +  /7-Fetc.  _a  _c       eg 

h  +  ,/  +/-f  h  -h  etc.  ~  '  ~  b~  d-'f-  h~  ^^^'  ^'^'^• 

(Ax.  4) 

a.   Continued  proportions  are  often  written  more  compactly,  thus, 
a  :c  :e  :  lb  :d  :  f  instead  of  a  :  6  :  :  c  :  (Z  :  :  e  :/. 

The  former  is  therefore  not  one,  but  two  different  proportions. 

6.  The  above  theorem  finds  its  most  important  application  in 
geometry,  where  by  means  of  it  the  perimeters  of  similar  polygons 
are  found  to  be  proportional  to  any  two  homologous  sides. 

2.  Tke  correspondltig  ten/is  of  two  or  more  proportions 
may  be  multiplied  together,  giving  rise  to  a  new  proportion 
which  holds  If  the  given  proportions  Jwld, 

If  a:b::c:d 

a'.b'.'.c'.d' 
a"'.b":'.c"'.d" 
then  aa'a"  :  bb'b" : :  cc'c" :  dd'd!' 

For,  writing  the  given  proportions  in  the  equation  form, 
and  multiplying  them  member  by  member, 

a c      a!        c'       a"       c" 

aa'a"        cc'cf' 


bb'b"       dd'd' 


Q.  E.  D.  (Ax.  3,  178) 


and  so  for  any  number  of  proportions. 

Moreover,  it  is  evident  that  the  terms  of  two  proportions 
may  be  divided.     Thus,  if 


r-^-'^y 

df 

a    b      c    d 
a'''b'''''^'d'' 

390  TEXT-BOOK  OF  ALGEBRA.  • 

378.  Special  Forms  in  Proportion.  — 

1.  A  7)iean  py^oportlonal  between  two  quantities  is  one 
whose  square  is  equal  to  their  product. 

Thus^  if  a:b::b\c,  h  is  a  mean  proportional  between  a 
and  c. 

For  by  the  first  theorem,  h'^  =  ac  (or  b  =  ^ac). 

The  third  quantity  e  is  called  the  third  ■proportional  to  a 
and  b. 

a.  A  mean  proportional  is  not  at  all  the  same  as  an  arithmetical 
mean.  Thus,  the  arithmetical  mean  of  6  and  4  is  5,  or  half  their 
sum;  while  the  geometrical  mean,  the  mean  proportional,  equals 
\/6  X  4  =  V2A,  which  is  manifestly  less  than  5. 

2.  Two  mean  proportionals.  —  If  a  \b::b:G::G:d,  b  and 
c  are  two  mean  proportionals  between  a,  and  d. 

Prove  (1)    a'.G::a^:b^     (2)    a:d::  a^ :  P 

379.  Exercises  in  Ratio  and  Proportion. 

1.  Which  is  the  greater  ratio,  3  :  4  or  ,3^ :  4^  ? 

2.  Write  the  ratio  compounded  of  8  :  7  and  5 :  6.    Which 
is  less  and  which  is  greater  than  the  compounded  ratio  ? 

3.  The  last  three  terms  of  a  proportion  being  4,  6,  and  S, 
what  is  the  first  term  ? 

4.  Find  a  third  proportional  to  25  and  400. 

5.  Find  a  mean  proportional  between  2f  and  /\j. 

6.  Arrange  in  order  of   magnitude  4:3;  9  :  <S  ;  2;") :  23, 
and  15: 14: 

7.  If  the  ratio  of  a  to  b  is  2p5  what  is  the  ratio  of  3  a  to 
4.b? 

8.  Prove  that  if  a  :b  ::  c:  d  that  a  —  h  \  r  —  d  ::  a  -\-  b: 
c  -^d. 

9.  If  the  ratio  of  m  to  n  is  i,  what  is  the  ratio  of  m  —  n 

to  7?1  +  7i  ? 

10.  Why  is  a"^  —  b'^  the  meaii  proportional  between  a^  -\- 
2  ah  +^2  and  a^  _  2  ah  +  ^^9 


Toi'ir 


i;i:i..\  11. i>    i<  >   i-.<  >i  A  ri(  »ns.  -W)! 


11.  If  a  :b::  rid  prove  that  ra^  :sb^::  rc^  :  sd\ 

12.  Find  two  luiiubers  in  the  ratio  of  .S :  4,  (suggestion 
I]  X  and  4r)  of  which  tlicir  sum  is  to  tlie  sum  of  their 
sfjuares  as  7  to  ;">n. 

13.  Solve  X'  —  Aix-  —  \)::x-  —  .">  j-  +  (> :  x'  -\-  -i  x  -\-  'X 

14.  Solve  for  x  in  the  proi^rtion  (a  —  x)  :  (x  —  ^,)  :: ,,  -,  /,. 
16.    Solve  (1)    x:  f/::3:5     (2)    ar :  4  : :  15  :  t/. 

16.  \\  hat  (juantity  must  be  added  to  each  of  the  terms 
of  the  ratio  in  :  n  so  that  it  may  become  equal  to  that  of  j^ :  q  ? 

17.  Four  given  numbers  are  represented  by  ?w,  ii,  j),  and  q ; 
what  (piantity  added  to  each  will  make  them  proportional  ? 

18.  If  four  quantities  are  already  proportional,  show  that 
there  is  no  number  which  \mng  added  to  each  will  leave  the 
resulting  four  numbers  proportional. 

19.  If  a:  ft : :  f) :  c  prove  that  a  -{-  h  :  h  -{-  r  ■ :  a  :  h. 

20.  \i  a\h\\c:d  prove  that  3  a  -\.h\h\ :  3  f  -\-  d  :  d. 

21.  \i  a:h:.c\d  jirove  that  2a-\-^h.'^a  -  4  /> : :  2  ^  + 
.3  ,Z :  3  c  -  4  </. 

22.  U  {a^h-]-c-\-  d)  (o  -b-c-\-d)  =  {<i  -h-irc-  d) 
(a  -\-h  —  c  —  d)  prove  that  a  \h  ::  r  .d. 

23.  If  </:/>::  ^' :  ^'Z  is   {mn  -j-  itl>  i  :  (  //"  ^^  qb)  : :  {mr  -[-  nd)  : 

{pc  -t-  qd)  ? 

24.  If  a:h\'.\\  o,  d  :/::  r,  :  _'.  -  .  . :  C,  ;  7,  d  :b::7  :  .'5, 
and/:  r : :  4 : 3  to  what  iuunl»ti  -  i>  t  lie  continued  ratio  a  :  h : 
*':d:e::  proportional  ? 

25.  The  product  of  two  numbers  is  112,  and  the  differ- 
ence of  their  cubes  is  to  the  cul)e  of  their  difference  as 
31  :  3.     What  are  the  numbers  ? 

26.  Given  Va;  -j-  a  :  y/x  —  a::m:  n  to  find  x. 

27.  Find  two  numbers  whose  sum,  difference,  and  product 
are  proportional  to  m.  w,  and  />. 


392  TEXT-BOOK  OF  ALGEBRA. 

380.  Variation.  —  One  quantity  is  said  to  vary  with 
another  when  there  is  some  law  connecting  changes  in 
their  values.  If  3/  is  a  function  of  x,  then  as  x  changes 
in  valne  y  also  changes.  The  simplest  Cases  of  variation 
(the  only  ones  the  word  as  used  technically  in  algebra 
refers  to)  are  closely  connected  with  ratio  and  proportion. 
Indeed,  direct  variation  is  nothing  other  than  a  new  aspect 
of  proportion. 

1.  Direct  Variation.  —  One  quantity  varies  directly 
with  another  when  as  one  changes  the  other  changes  in  the 
same  proportion. 

Thus,  the  distance  a  man  travels  in  a  day  varies  as  the 
number  of  miles  he  travels.  If  he  travel  5  miles  per  hour 
he  will  go  over  |  times  as  much  ground  as  when  he  travels 
only  3  miles  per  hour ;  and  if  he  travel  8  hours  per  day,  he 
will  journey  only  |  as  far  as  when  he  travels  10  hours  per 
day. 

When  one  quantity  varies  directly  as  another,  or  simply 
varies  as  another,  they  are  to  each  other  in  a  constant  ratio. 
This  is  involved  in  the  definition. 

For,  if  a  varies  as  h,  and  a'  and  h' ,  a"  and  h" ,  etc.,  are 

other  values  of  a  and  h,  then  _==  —  ==  —^  =  etc, ;  i.e.,  the 

b        b         b 

ratio  is  the  same  in  each  case,  or  is  constant.  When  one 
quantity  varies  jointly  as  each  of  two  (or  more)  others,  as 
in  the  example  given  above,  it  varies  as  their  product. 

As  an  illustration :  if  one  train  runs  30  miles  per  hour 
and  22  hours  per  day,  while  another  train  runs  25  miles  per 
hour,  and  23  hours  per  day,  the  distances  they  will  pass 
over  in  any  given  time  are  proportional  to  the  'products  of 
these  numbers,  or  the  rate  and  time. 

2.  Inverse  Variation.  —  One  quantity  varies  inversely 
as  another  when  as  one  increases  the  other  decreases,  so 
that  their  product  is  constant. 


ToiMrs    IIKI.ATEI)   TO    KQUATIONS.  393 

Tims,  if  5  iiu'ii  can  do  a  job  of  work  in  16  days,  10  men 
can  do  the  same  work  in  8  days,  20  men  in  4  days,  and  so  on. 

If  a  vary  inversely  as  b,  and  a'  and  b'  are  corresixjnding 
values,  then  a:  a!  \\  b' :  b,  since  this  giv'es  ab  =  a'b'  which 
is  true  by  definition. 

If  one  quantity  varies  directly  as  a  second  and  inversely 
as  a  tliird,  it  varies  as  the  quotient  of  the  second  dividinl 
by  the  third. 

Thus  the  number  of  bushels  of  wheat  elevated  in  a  mill 
varies  directly  as  the  amount  of  work  i)erformed,  and  in- 
versely as  the  height  to  which*  it  is  raised. 

3.  Other  Kinds  of  Variation.  —  One  quantity  may 
vary  as  any  function  of  another.  It  is  shown  in  mechanics 
that  if  5  is  the  distance  in  feet  which  a  body  falls  in  t  sec- 
onds, then  s  =  16  f^.  In  other  words,  the  distance  varies 
directly  as  the  square  of  the  time.  Xewton's  law  of  gravi- 
tation states  that  the  attraction  of  the  heavenly  bodies  for 
each  other  varies  inversely  as  the  square  of  the  distance. 
Works  on  elementary  algebra  usually  confine  their  state, 
ments  and  exercises  to  the  first  two  kinds  of  variation. 

4.  ExERriSK. 

(1.  If  X  x.  II  and  //  =  5  when  a*  =  14  liiid  ./•  when  y 
=  20.  *  '  • 

(2.  \i  X  X  y  and  a;  =  7  when  ?/  =  .'U  find  y  when  x 
=  20. 

(3.  If  ^  oc  i?Cand  ^  =  6  wlien  //  =  4  and  C  =  15, 
find  B  when  A  =  100  and  C  =  10. 

(4.  If  7n  QC  i,  and  when  in  =  >/.  n  =  />.  find  ;/  when  m 
=  c. 

(5.  If  the  cube  of  x  varies  as  the  scjuare  of  y,  and  if  x 
=  3  when  y  equals  5,  find  the  equation  between  x  and  y. 

(6.  If  a  -\-  b  cc  a  —  b  prove  that  «^  -)-  b'^  x  «/»;  and  if 
a  cc  b  prove  that  a^  —  b^  x  ab. 

SrooESTioN.  —Let  m  =  the  common  ratio,  then  in  the  first  part 

a  +  6  =  m  (a  —  6). 


394  TEXT-BOOK    OF   ALGEBRA. 


CHAPTER    XX Vr. 

EXPONENTIAL  E(,)UATrOX8.  — LOGARITHMS. 

381.  Exponential  Equations  are  those  in  which  the  un- 
known appears  as  an  exponent. 

Tims,  5^'  =  25,  in  wliieli  plainly,  x  =  2. 

8-^  =  500,  in  which  plainly,  ic  >  3  and  <  4. 
a^  =  Z*  is  the  general  form  of  the  equation. 

a.  Except  in  the  case  of  exact  powers,  such  as  2-*  =  16  (in  which 
•^  —  4);  (1)^  =  \  (in  which  a:  =  |);  etc.,  there  is  no  process  like  those 
for  the  solution  of  simple  and  quadratic  equations  by  which  this  class 
of  problems  can  be  solved.  However,  tables  of  logarithms  enable  us 
to  find  approximate  solutions  very  readily.  We  turn,  therefore,  to  the 
subject  of  logarithms. 

382.  Tlie  Logarithm  of  a  number  is  the  exponent  of  tlie 
power  to  which  a  fixed  number  must  be  raised  to  produce 
it.     The  fixed  number  is  called  tlie  base. 

a.  Evidently  three  different  numbers  are  concerned.  First,  the 
fixed  number;  second,  any  number;  third,  its  logarithm. 

Thus,  if  the  base  is  10,  and  the  number  105,  its  logarithm  is  2,0212, 
i.e.,  a  little  more  than  2.  For  10'^  =  100,  and  10^""^'  =  105.  And 
so  for  any  other  number.  As  another  example,  the  logarithm  of  865 
is  2.9o70  nearly. 

b.  Logarithms,  except  in  a  relatively  small  number  of  instances, 
are  found  approxiniatchj  and  not  exurtly. 

Thus,  the  logarithm  of  105  is  not  exactly  2.0212,  in  which  four 
decimal  places  of  its  value  are  given,  nor  is  it  2.021180,  in  which 
six  places  of  its  value  are  given,  but  these  are  approximations  cor- 
rect as  far  as  they  go. 


TOPICS    lM:i,\ri.I»    TO    KQUATIONS. 


nil") 


383.  Systems  of  Logarithms.  Tables.  Any  positive  nuiii- 
Ijer  t'xcept  unity  may  lu*  taken  as  tho  baso  of  a  system  of 
logaritlims. 

<i.  The  logarithm  of  1  in  any  system  is  0;  for  any  nunilM^r  (tlic 
basf)  whose  exponent  is  zero  is  equal  to  unity.     (128,  1.) 

.1.  Let  us  take  2  as  the  base  of  a  system.  The  numbers 
are  written  in  the  left  column  and  directly  opposite  them 
their  logarithms. 


NUMBER. 

lo<;a- 

KITllM. 

nEASON. 

NUMBER. 

LOGA- 
RITHM. 

REASON. 

1 

2 

i=.5 
3 
i=.33  + 

0. 

1. 

1.5^50 
-1.5850 

2«         =1 

2^          =2 
2-        =i 

O1-6SC0       O 

2-'""  =  J 

4 

i=.25 
5 

^=.2 
etc. 

2. 

-2. 

2..3223 
-2.3223 

etc. 

2»          =4 
2-»       =i 

93.3SS3     K 

2-».38a3_  1 

etc. 

The  student  is  not  supj)osed  to  know  how  the  decimal 
logarithms  are  found.  One  method  of  obtaining  them  is 
by  a  long  process  of  repeated  extraction  of  roots.  See 
the  "  Encyclopajdia  Britannica"  article  on  Logarithms. 

h.  Logarithms  are  exponents,  integral  or  fractional,  and  as  such 
may  be  inteqireted  as  all  others.     (283.) 

Thiis,  2''""  =21*888^  means  the  1.5850th  power  of  the  10000th 
root  of  2;  and  if  these  operations  were  actually  performed  the  result 
would  be  3. 

So,  also,  10 ■""'"  =  10>2i*88o^  means  the  21189th  power  of  thr 
l(XX)000th  root  of  10;  and  if  these  operations  were  actually  perfoniuMl 
the  result  wouhl  be  1.05.  In  other  words,  the  logaritlfin  of  1.05  in 
the  system  whose  base  is  10  is  .021189  correct  to  six  <le('iinal  plact  s. 
So  for  all  logarithms. 

2.  lenity  cannot  be  tlie  ])ase  of  a  system  of  logarithms, 
for  every  power  of  1  is  1  continually.  Neither  can  7iegntlrt> 
numl>ers  he  used  as  the  base  of  a  system  of  real  logarithms  ; 
for  we  saw  in  this  article  that  the  root  to  be  extracted  is 
even.     Hence,  this  root  would  be  imaginary.     Then  when 


396  TEXT-BOOK   OF    ALGEBRA. 

raised  to  the  various  powers  giving  the  corresponding  num- 
bers, some  would  be  imaginary,  some  plus,  some  minus,  and 
all  in  confusion.  Therefore,  a  minus  number  cannot  be  the 
base  of  a  simple  system  of  logarithms. 

c.  The  base  of  a  system  being  taken  positive  and  greater  than 
unity,  the  logarithms  of  all  fractions  between  0  and  1  are  negative 
(see  table  in  this  article) ;  the  logarithms  between  1  and  the  base  of 
the  system  are  between  0  and  1;  and  the  logarithms  of  numbers 
greater  than  the  base  are  greater  than  1.  Negative  numbers,  as  such, 
are  not  supposed  to  have  logarithms,  though  numerical  calculations 
in  which  negative  numbers  occur  are  often  made  irrespective  of 
signs,  or  just  as  if  all  were  positive. 

384.  Terms  and  Notation.  The  integral  part  of  a  log- 
arithm is  called  the  characteristic.  The  fractional  part  is 
called  the  mantissa.  "  Logarithm  "  is  usually  abbreviated 
into  "log.,"  and  the  base  of  the  system  used  may  be  written 
as  a  subscript  to  it. 

Thus,  log.  looO  ==  1.6990  means  that  the  logarithm  of  50 
to  base  10  is  equal  to  1.6990.  In  this  logarithm,  1  is  the 
characteristic,  and  .6990  the  mantissa. 

385.  The  Briggslan  or  Common  System  of  Logarithms.^    The 

system  in  practical  use,  like  the  Arabic  notation,  has  10  for 
its  base. 

1.  Logarithms  of  fractions  in  this  system  (383,  c)  are 
negative,  and  grow  larger  as  the  number  grows  smaller. 
Thus  the  logarithm  of  .001  is  -  3 ;  that  of  .0001  is  —  4, 
and  so  on  ad  infinitum.  At  the  limit  when  the  number  is 
infinitely  small  (^^  0),  the  logarithm  is  an  infinitely  large 
negative  number.  Furthermore,  it  may  be  seen  that  the 
logarithm  of  a  fraction  say  between  .001  and  .0001  must 
lie  between  —  3  and  —  4,  one  between  .0001  and  .00001 
between  —  4  and  —  5,  dependent  on  the  powers  of  -^^  ;  and 
so  on. 

1  Logarithms  were  invented  by  Lord  Napier  about  1614  A.D.  Henry  Briggs,  a 
great  admirer  of  Napier's  logarithms,  saw  the  practical  advantage  of  using  10  as 
the  base,  and  constructed  his  tables  accordingly. 


KM'K  >   i;i:i..\  ii;i)    ro   i:«.»r.\Tin\s.  397 

2.  Logarithms  between  1  and  10.  —  The  h)garithm  of  1 
is  0,  and  of  10  is  1 ;  hence  the  logarithms  of  numbers 
between  1  and  10  lie  between  0  and  1,  i.e.,  are  proper 
fractions. 

3.  Logarithms  of  numbers  greater  than  10.  We  have  10^ 
=  10,  10'^  =  KM),  10 «  =  1000,  10*  =  10000,  10*^  =  100000, 
10«  =  KXMHKM),  etc. 

Here  we  may  see  that  the  logarithm  of  a  number  between 
10  and  100  is  between  1  and  2 ;  the  logarithm  of  a  number 
between  100  and  1000  is  between  2  and  3,  i.e.,  is  2  -|-  a  deci- 
mal, and  so  on,  ad  injinitum.  Hence,  10*  =  oo  .  Here  as 
elsewhere  the  two  x  's  are  not  equal. 

To  further  illustrate  the  preceding, 

log.  07  =  1.8261  log.  345.25    =  2.5381 

log.  1  ( )r,0.45<)  =  3.0292  log.      5.678  =  0.7542 

386.  Briggsian  Mantissas.  —  In  the  Briggsian  or  common 
system  tlir  rnluc  of  tin-  M.VNTissA  wUl  uot  chduge  if  the  num- 
ber be  multiplied  or  divided  by  some  power  of  10.  In  other 
words,  the  value  of  the  mantissa  is  indej)endent  of  the 
position  of  the  decimal  point. 

Thus,  log.      1.27  =  0.1038 
log.      12.7  =  1.1038 
log.   12700  =  4.1038 
etc.  etc. 

The  reason  for  this 'follows  from  the  law  of  exponents. 
Let  us  supi)ose  that  the  logarithm  of  1.27  is  known  t(>  be 
0.1038,  or  that  lO'*-^^'^  =  1.27.     We  have,  (287.  2) 

(1)  12.7=  1.27    X    10  =   10^1088  X    1(>1  =   1()0.108H+1  _  1()1.1088 

(2)  .(H27  =  1.27  -^  \m  =  lo^^****  -^  W'=  l()o  »o»8-^  =  |(,iio88 

(3)  1270=  1.27  X  1000=  10«»0-^8  X   10'  =    l()0.108».+8  _   |()8.I0«« 

(4)  .(MM)127   =    1.27    H-    1000()   =    1()0  108«    ^   1(>4_    J^(,0.1088-4 

__   j()4.1088 


398 


TKXr-nooK    OF    ALGKI'.ILV. 


6   ju    7    M   8    D    9 


0212  4110253  41  0204  +»  o;J34i40 


15  1701  bi 

16  2041  ■>: 

17  2304  2.; 

18  {2553!24 

19  12788  d 


1790  H 1818 
2008  27:2095 
2330 


2577 

2810 


25: 2355 
24!  2601 
23{2833 


29j  1847 

27;  2122 

25  2380 
24  2(525 

23  2856 


:^^8,1875  2J 

2148,2; 
2405  2. 
2648 '2. 
2878  22 


0607  -■»{  0645  ;37  0682 
09()9'a5{1004|34  1038 
1303  32!  1335:32  1367 
1614 :3o|  1644  29  1673 


1903 


»,  1931  i28  1950 
217512612201  20  2227 

25!  2480 


2430  25 

2672  23 
2900 123 


2455 
2695 
2923 


23  2718 
2945 


0719j36 
1072134 
1399  b 
1703  29 


0374140 
0755 137 
il06 
1430 
1732 


2811987  27  2014 
26|2253  26  2279125 

24  2504  25 


M-2742 
2967 


2529124 

2765  !23 
2989 '21 


20  3010  22 

21  ;3222!2i 

22  ^•U24'2(. 

23  3()17  i;i 

24  ,  3802  i  18 


3032 
3243 
3444 

3636 

3820 


26 

27 
28 
29 


30 

31 
32 
33 
34 


3979 
4150 
4314 
4472 
4624 


3054 
3263 
3464 

19  3655 
18  3838 


21!  3075  21 
3284,20 
348319 
3674] 

3856^18 


3096  !2i 

3304 

3502 

3692 

3874 


3118 
3324 

3522 
3711 

3892 


3139  21 
3345 120 
3541 
3729 
3909  18 


3160 
3365 
3560 
3747 

3927  18 


3181 
3385 
3579 
3766 
3945 


20  3201 
19  3404 
19  3598 

18  3784 
17  3962 


18  3997  17  4014  |i7  4031 
16  4166  17  4183  17  4200 
10  4330  10;  4346  1.;  4:562 


15  4487 
15  4639 


15  4502  jioi  4518 
15  4654  id  4669 


i7  4048ji7 
164216  16 


4378 
4533 
4683 


477 1115  4786  :i4|  4800 
4914!i4  4928  14  4942 


5051  14  5065  14 
5185  13  519813 
531513  5328,1 


5079 
5211 
5340 


1414814 
'4955 


35  5441  12 

36  55(53 1 12 

37  5(582  1 
38 
39 


5708 
5911 


5453  12  5465  13  5478  12 
5575  12  5587 112  5599  12 
5694  11  5705 1 12  5717  12 
5809  12  5821  115832  11 
5922  11  5933  111  5944  11 

40  (5021  10  6031  iij  (3042  |ii  (5053  11 

41  (5128  10(5138  11  (5149  |n;  6160  10 

42  6232  u  6243  10  6253  10  6263 1 


5092 

13  5224 

5353 


15  4829 
4969 
5105 
5237 

13  5366 


6335 
6435 


10,634510 
916444  10 


6355 
6454 


10  6365 1(» 
10  6464 10 


5490 
5611 
5729 
5843 
5955 


6064 
6170 
6274 
6375 
6474 


4065 
4232 
4393 

4548 
4698 


4843 

4983 
5119 
5250 

5378 


5502 
5623 
5740 

5855 
5966 


4082  ii7 
4249 116 
4409  16 
4564  15 
4713  15 


4099  |i 
4265  16 
4425  15 
4579  15 
4728  !i4 


4116  i7j  4133 

4281  17  4298' 
4440  16  4456 


4594  Ii5 
474215 


4857 
4997 
5132 
5263 
5391 


5514 
5635 
5752 
5866 
5977 


6075 
6180 

6284 
6385 
(M84 


6085 
6191 
6294 
6395 
6493 


4871 
5011 
5145 
5276 
5403 


5527 
5647 
5763 

5877 
5988 


4886 
5024 
5159 
5289 
5416 


12,5539 

115658 

12  5775 
11;  5888 
11  599!) 


4609 

4757 


4900 
5038 
5172 
5302 

5428 


5551 
5670 

5786 
5890 

(ioin 


116096' 
10  6201 
10  6304 
106405 
10  6503 


,10(5117 
10  (3222 


11  610 

11  6212 

10  6314 

10  6415  iio  6425 

10  6513  9  6522 


6325 


45:653210 
46  1(3628 


6721 

(3812 
6902 


50 

(3990 

51 

7076 

52 

7160 

53 

7243 

6542 
6637 
6730 
(3821 
6911 


6551 
6646 
6739 

6830 
6920 


10  6561 
10(3656 
10  6749 

9  6839 
816928 


6571|9 
6665  10 

(3758  jo 
684819 
6937!  9 


6580  10  6590 

(567519 '(5684 
67(57  !>  6776 
()857i9  (58(36 
6946 1 9i  6955 


9:6599 
9 1 6693 

9  6785 
9!  (5875 
9 1 6964 


54    7;;24 


8  6998'  9 
708419 
7168  9 
7251  Is 
7332:8 


7007  9 
7093  8 

717718 
7259  .'^ 
7340  s 


7016 
7101 

7185 
72()7 

7;ns 


7024  9 
71108 
7193  0 
7275  9 
735(5  s 


703319  7042 
7II818  7126 
7202  8^7210 

7284  8:7292 
73(54  8  7372 


6609 
6702 
6794 

6884 
6972 


6712 
6803 
6893 
6981 


{8.17050 
9 1 7135 
817218 
8 1 7300 
|8!  7380!  s' 7388 1817396 


917059  8  j  7067 
87143  917152 
817226  9 '7235 
S7308  8 1 7316 


ATIO^S. 


3119 


N     O    1.    1    >.    2    I.    3    i»    4    ■> 

5    1)    6    I)    7    1'    8    i>    9    1) 

55  7404  s  7412;7  7419  «  7427,H;74;jr) 

8 

7443  8  7451  «  7459  r 

74(50  8  7474  « 

56  i7482's!74{M)  7  7497  8|7')()5 

751:5 

7 

7520.8 

7528 ;«  75:50,7 

7543 18  j  7551  8 

57  1 7551)  7  750<5  s  7574  8  { 7582 

7580 

8 

7597  7 

700418 

7012 j 7 

i7010|8 

7(527  7 

58  i7G;U  ^'7042  r  7<{4()«, 7(557 

7004 

8 

7072 

7 

7070  7 

7080 j 8 

7(594 1 7 

7701  8 

59    77o«»  :  771  ;  ;  772:1 18  77;il 

77:J8!7 

7745 

7 

7752  is 

V700!- 

I7707J7 

7774 18 

00.77.S2  r  77s'.i  :  770(5  7 

7803 

7810!8 

7818 

7 

7825 

7!  78:32 '7 

j  78.30 

7 

7840[7 

61  i 785.1,7, 7800 |H  7808  7 

7875 

7882 j 7 

7880 

7 

7800 

7  7003  7 

i7010 

7 

7017 1 7 

62    7024  7!70:ir7  71KJ8:7 

7045 

70521? 

7050 

7 

70(50 

7!  7973 17 

'7080 

7 

7087  « 

63    IW.]  7  8000  7  S007  7 

8014 

80211 7 

8028 

7 

80:l5i«8041  7 

8048 17 18055 '7 

64    8i)t52  r  80()i>  (i  8075  i  7 

8082 

80801 7 

8000 

Gi8102;7   8100   7i8110i6j8122:7 

«5i8120,7  8i:J(5«|8142  718149 

8150 

« 

8102 

7 

8100  7 

8170JC 

8182|7J8180  <•. 

66  i 8105  7  8202  7  8209  6I8215 

8222 

6 

8228 

7 

8235  fi 

8241 1 7 

8248  «  8254  7 

67    H2(51  c;82(57  7  8274  0  8280 

8287 

6 

h-j9;5 

t; 

^299  7 

8300  c 

8312!7!8310  0 

68   h:]25  c: 83.31 '718; J;{S  «  8:544 

8351 

6 

KW,1 

,; 

.s:;(;3  7 

8:J70  ( 

8:170  c' 8.382  « 

69  .8:188  7  8;}95;fl'8401  j«;8407!7 

8414 

6 

8420 

1; 

8420:6 

84:52  i  7 

84.30  fii 8445 :« 

70!845l|fl'8ir,7l«!84(5.3 

7|8470|6|8470 

6 

8482!«,8488ic 

8404 1 « j  8500  C:  8.506 1 7 

71    851 •                  ^525 

(i 

8531  6!  85:57 

6 

8543j«|8549j6 

8555  |r.i  8501  «  8567!fi 

72   m:,7.;                --)85 

6 

8591  .;  8597 

6 

8(503|6  8000:g 

8015  c 

18(521  «  802710 

73       •■            ..;»  .;  ?S(>45 

6 

8(551 16!  8(J57 

6 

8(5(53  6  8000 in 

8(575  6 

j8081  5  8(58(5  i  6 

74                   ■.'.i8i6J8704 

6 

87106|8710 

6 

8722  5!8727|6 

87.33  fi 

! 87:10  c  8745ic 

75  >:01    •  >7.')»5  <!'87'»2  6  87(58 j 6 

8774 

5 

8770 

6  8785|6|870l|6  8707 ;s!8802ifi 

76    ><^os  •;  ,ssi4  <>  SS20  5  8825  0 

8831 

6 

88:57 

A 

8842 i6  88481 « 

88.54  i  51 8850^6 

77    HS«i.')  .:  ,sH7I  .',  XH~()  c;8H82  r,  8887 

6 

889:l 

6  8800i«;8904  (i  8010  518015  « 

78   W2\  »;  H<>27  .'.  H«»;52  fj  8!>:I8  5  894:5 

6 

8040 

5,8954  6;89(;o  .',  8905  6  8071-5 

79    s.iT,:  .:  .'ts-i  .,  8987 |6i 89931 5! 8998 

6 

0004 

fil (MKM)  6|0()15 15  9020 i«| 0025  6 

HU                   '  ;t5  <:  9042 

« 

0047 

6  {)053|« 

0058 

« 

0003 

6 

0060  5 

0074  6 1 0070!  6 

81    .-'-              '     !>09f5 

0101 

5  0106 16 

0112 

a 

0117 

5 

0122  6 

0128  .'i!oi:33:.'. 

82   913-                   1149 

0154 

5  0150  6 

0105 

5!  0170 

5 

0175  5 

9180  <;  918(5  :, 

83   0191      ...  .  1)201 

0206 

6  0212  5 

0217 

5  0222 

5 

0227  5 

92.32 i«| 92:18  .'. 

84  9243,5  9248 15! V>25:} 

0258 

5  02(53  e 

0200 

5I0274 

5  0279 1 5 

0284  |5i  9280  i.i 

85  9294  .1  i  9290  5  i  0304 ' '. !  0:J00 

6  0315  5 

0320 

«|  0.325 

-J  1 0.3.30 1 5 

;9.3.35!5  0340  .'. 

86    9.545  V0350  .-.  9355  .■;  9:5(50 

5  9:5(55  5 

0370 

5  9375 

5  9:580  5 

9:585:.';  9:590  .-. 

87    9:59.'.  .  '.t»n;>  -.  .141).-,  .-.  04 10 

6  9415  5 

0420 

.'.  9425 

5!  94.30  5 

94:55  .5  9440  .-. 

88    944'.                     155  .5  94(50 

5  94r,5  4 

94 '!9 

-.  !I474'.'.  9479  5 

948415  948915 

89    9491                      504 15  950914  9513  5 

95  IS 

.',9523  .',,9528  5 

95:53 15  95.38  4 

J><)  i>54-                    '552  5  9557!*  05(52 

4 

9.500 

5|  0.571 1«|  0570 1 5 

0.581,.'.; 9.580  4 

91    959n                     »'A)0  •>  mo'y  i  9«}09 

5 

0(514 

A  <m;i<»  a  <m\24  4 

9(528  5  00.33  5 

92    IMJ:;-                   .,;47  .-,  «>(;.-,2  ■-.  9*557 

4 

iM5(;i 

•hI  4 

<»(;75  .-.  9(580  .V 

93    iM58.,                   Mi04  .'.  iM5<»1>  4  970:} 

.5 

07(^^ 

•  717  :,  9722  .'.  0727  4 

94  ; 07:51                    '741 14, 07451 5, 0750  4 

0754 

.-4  97  59 ;  4 1 9703  i  6 1 0708  r,  \  »7 73  4 

95 '  077 7                   t780 ! '.  0701  <  0705  a 

08<M> 

'  «»s(i.-.'4'u800  5 

9814 '4  9818  '. 

96   98_>;                   18:52  4 

08:K5  .'J0841  4 

9S4.-, 

'■>:,4  r. 

985<>  4  98(53  .'. 

97    98«;s                  i.s77  4 

0881  5!  0886  4 

WShi 

1        .•>.9!>  4 

990;5  .V9908  * 

98    9912  .',  9917  4  1>021  .-. 

SK>2(5  4  IKKIO 

4 

oo:i4 

5  5^939; 4, 5)043  5 

9948  4  9952  4 

99    JH)5<5  5  00(51  4  •HX55  4 

1KW59  -.  0074 

4 

9978 

.5  1)08:1  4  0087  4 

9iM)i  .'.  oihm;  4 

400  TEXT-BOOK   OF   ALGEBKA, 

a.  Mantissas  are  always  taken  positlvel)/.  Otherwise  they  would 
change  when  the  characteristic  became  negative.  Thus  the  loga- 
rithm of  .127  is  0.1038  — 1  = —0.8962.  But  by  considering  the 
characteristic  alone  as  negative,  we  avoid  the  introduction  of  a  new 
decimal.  In  such  cases  the  minus  sign  is  written  over  the  character- 
istic^ and  not  before  the  logarithm. 

Thus,  4.1038  means  the  binomial  —  4  +  .1038  =  —  (3.8962). 

387.  Explanation  of  the  Accompanying  Four-Place  Table. ^  — 

The  table  gives  tJis  mantissas  only  of  all  numbers  from  100 
to  999.  The  first  two  figures  of  the  numbers  are  found  in. 
the  column  marked  "X,"  the  third  is  one  of  the  ten  figures 
at  the  top  of  the  page. 

Thus  487  is  found  by  taking  the  48  in  the  ''  N  "  column, 
and  the  7  from  the  upper  row  of  figures.  The  mantissa  of 
the  logarithm  of  487  is  found  in  the  7  column,  opposite  48, 
and  is  .6875,  At  the  intersection  of  lines  and  columns  are 
found  the  900  mantissas  corresponding  to  the  900  numbers, 
tlie  tw^o  first  figures  determining  the  line,  and  the  last  the 
column.    A  decimal  point  before  each  mantissa  is  understood. 

The  columns  of  numbers  marked  D  are  merely  the  differ- 
ences between  the  mantissas.  Thus  the  difference  between 
.31)32  and  .3054  is  22 ;  and  between  .4133  (corresponding  to 
259)  and  4150  (corresponding  to  260)  is  17. 

It  may  be  added  (see  386)  that  this  table  gives  also  the 
mantissas  of  all  numbers  consisting  of  three  or  less  than 
three  figures,  preceded  or  folio ired,  hi/  any  number  of  ciphers, 
(I  lid  Irrespective  of  the  posit  ion  of  the  decimal  point.  Thus 
t'.e  mantissa  of  31  is  the  same  as  that  of  310,  and  is  .4914; 
t'le  mantissa  of  8  is  the  same  as  that  of  800,  and  is  .9031. 

388.  To  Find  the  Logarithms  of  Numbers. 

1.    To  find  the  characteristic  of  the  logarithm  of  a  number. 

(1.  Numbers  greater  than  1. 
(1)  The    logarithms    of   all    numbers    consisting   of   one 

1  riu' six-place  tables  prepared  by  Professor  (i.  W.  Jones,  of  Cornell  University, 
Iliiaca,  \.V.,  are  among  the  be^t  of  tliosc  imblisbcd  in  tliis  coiuifrv. 


T()i'i(  s   i;i:latei)  tu  KgUAiioNs.  401 

integml  fi«,'ure  (i.e.,  between  1  and  10)   liave  0  for  <a  ehar- 
actt'ristie,  since  they  must  lie  In^wj-en  0  juul  1.     (385,  2.) 

Thus  2,  5,  4.75,  6.978,  9.<S'.)  I.;,  dr..  all  have  (I  tor  the  char- 
aeteiistic  of  their  h)garithnis. 

(13)  Tlie  h)garitlims  of  all  numbers  l)etween  10  and  KM), 
consisting  of  two  figures  before  the  decimal  point,  have  1 
for  a  characteristic,  since  their  logarithms  are  all  between 
1  and  'J.     (385,  o.) 

Thus  40,  75,  35,  96,  89.746,  etc. 

(8)  The  logarithms  of  all  numbers  between  iOO  and  lOOO 
have  2  for  a  characteristic. 

Tho.se  between  1000  and  KMMM)  have  3,  and  so  on. 

The  chn  raff  eristic  of,  the  loijurithin  of  a  nnmher  [f  renter 
than  unitij  is  alirays  ofie  less  than  the  number  of  the  Jiyares 
lirrrrdinfj  the  decimal  point. 
(2.   Numbers  less  than  1. 

(1)  The  logarithms  of  all  numl)ers  between  1  and  .1 
(=<=  10"^)  have  —  1  for  a  characteristic,  since  the  values  are 
l>etween  0  and  —  1,  and  the  mantissa  is  always  added. 

Thus  .2,  .35,  .6978,  .934258,  all  have  -  1  as  characteristic. 

(2)  The  logarithms  of  all  numbers  between  .1  (=  !()-') 
and  .01  (=10-^)  have  —  2  for  their  characteristic,  since 
they  must  lie  between  -.  1  and  —  2,  and  the  mantissa  is 
added. 

63)  The  logarithms  of  all  nuinbcrs  between  .01  (=  10-^) 
;iii<l  .001  (=»  10-')  have  —  :;  I  i  ihtir  characteristic;  those 
l)etween  .(K)l  (=  lO"')  and  .oooi  (=  K)-*),  _  4,  and  so  <m. 

Thus,  for  .00()9  the  characteristic  is  —  3;  for  .0006972  it 
is  -  4 ;  for  .00002  it  is  —  5,  etc. 

The  characteristic,  of  the  lofjarithm  of  a  jtroper  fro rt ion  is 
always  neyative^  and  one  yreater  than  the  number  of  ciphers 
preceding  the  first  significant  figure  in  its  decimal  value. 

In  (1)  above  there  were  no  ciphers,  and  the  characteristic 
was  —  1.  Evidently  comnwn  fractions  must  be  reduced  to 
decimals  to  get  their  logarithms. 


402  TEXT-BOOK   OF   ALGEBRA. 

2.  To  find  the  mantissa  of  the  logarithm  of  any  number. 
The  rules  for  finding  the  characteristic  have  just  been  given. 
It  remains  to  investigate  the  method  for  finding  mantissas. 
(1.  The  finding  of  the  mantissas  of  logarithms  of  num- 
bers consisting  of  three,  or  less  than  three,  significant  figures 
was  explained  in  387. 

Write  for  practice  the  logarithms  of  the  following  : 


(1)  457 

(6)  40.0 

(11)  .0000679 

(16)  9000000 

(2)  345 

(7)  4 

(12)  3650 

(17)  21 

(3)  367 

(8)  259 

(13)  ^ 

(18)  371 

(4)  450 

(9)  .037 

(14)  .V 

(19)  2:1 

(5)  45 

(10)  .005 

(15)  520000 

(20)  1.11 

(2.  The  mantissas  of  numbers  consisting  of  more  than 
three  figures  are  found  by  interpolation. 

(1)  To  get  the  mantissa  of  a  number  consisting  of  four 
figures,  say  of  2568. 

By  Art.  386  the  mantissa  of  2568  equals  the  mantissa  of 
256.8.  The  reasoning  will  be  a  little  clearer  if  the  latter 
number  is  used.  Since  256.8  is  between  256  and  257  its 
logarithm  must  lie  (see  table)  between  2.4082  and  2.4099 
and  close  to  the  latter.  Now  one  would  naturally  suppose 
that  as  the  number  increased  from  256  to  257,  the  logarithm 
would  increase  proportionately,  or  at  least  nearly  so.  In 
truth,  it  does  not  increase  at  precisely  the  same  rate.  It  is 
shown  in  the  calculus  that  the  Briggsian  logarithm  of   a 

number,  n,  increases  ^ ^  times  as  fast  as   the   number 

n 

itself. 

Thus,  take  the  number  126,  and  suppose  it  to  receive  the 
increment   1  ;    then   the   increment   of    the    logarithm    is 

.     X  1  =  .0034  -f .    Adding  this  increment  to  the  loga- 

rithm  of  126,  which  is  2.1004,  the  sum   is  2.1038,  or  the 
logarithm  of  127.     But  in  passing  from  126  to  127  the  mul- 


loi'ics    i;i;i.Ari:h    to    Kt^UATluNiS.  40o 

4o4  -I-         434  4-  ^ 
tiplier  has  changed  from  to  — .- .^~.     Consequently 

the  aljovc   l()<4aiitliiiiic   iiiciviiu'iit  ().()()o4  -|-  ifc>  not  exactly 
cont'it,  but  sutticiently  so  tor  a  Four-place  table.     In  the 

exanii)le  before  us  the  rate  will  change  from  *  '  ^      to  *  '* 

which  is  a  very  small  difference.    Consecjuently  intermediate 
values  of  logarithms  may  be  found  by  proportion  with  only 
very  small  errors. 
We  have. 

Log.  2r)().    =  2.4(KS2  (Froui  table.) 

Log.  25H.8  =  2.4()1)()  (See  explanation.) 

Log.  257.    =  2.4099  (From  table.) 

To  interpolate  the  logarithm  of  2r>().«S  proi>ortionally  we 
say,  the  difference  of  the  numbers  :  its  fractional  i)art  = 
the  difference  of  the  logarithms  :  its  fractional  part. 

Here,  1  :  .8  =  17 :  a; .-.  a:  =  17  X  .8  =  13.f)  =  14—. 

(Imusmuch  iis  the  table  being  used  contains  4  decimal 
places  only,  14  is  taken  instead  of  18.(),  since  13.()  is  nearer 
to  14  than  to  13.) 

Adding  14  to  2.4082  gives  the  result  2.4096. 

(2)  For  a  greater  number  of  figures.  Thus,  to  find  the 
logarithm  of  52.7298. 

Log.  52.7        =  1.7218  (From  table.) 

Log.  52.7298  =  1.7220  (Ky  interpolation.) 

Log.  52.8        =  1.7226  (From  table.) 

Tabular  difference  =  8  ;  8  X  .298  =  2.384  =  2  +  : 

18  +  2  =  20. 
The  reasoning  is  the  same  as  before. 
From  these  examples  it  is  clear  that  to  find  the  mantissa 

»  In  order  that  this  change  may  be  «inall,  logurithinic  tables  begin  with  W)  or 
1000.  Four-phice  tables  include  the  numbers  from  100  to  V99.  Five  and  Six  place 
tables  have  four  figures  in  their  numbers,  and  extend  from  1000  to  9«9l». 


404  TEXT -BOOK    OF   ALGEBKA. 

of  a  number  consisting  of  more  than  three  figures,  proceed 
as  follows  :  — 

Find  the  mantissa  corresponding  to  the  first  three  signifi- 
cant figures  of  the  given  number.  Multiply  the  tabular 
difference  following  (column  D)  l)y  the  remaining  figures  of 
the  number  regarded  as  a  decinuil ;  and  add  the  product, 
taken  to  the  nearest  unit,  to  the  mantissa  of  the  first  three 
figures. 

Remaimv.  —  It  is  convenient  to  regard  the  differences  between  the 
mantissas  as  whole  muuberH  instead  of  retaining  their  decimal  places. 
No  mistakes  need  arise  in  so  doing. 

(o.    Write  for  practice  the  logarithms  of 

(1)  321.6       (4)  2.5675       (7)  .08742  (10)  .0004523 

(2)  14165      (5)  2379.6       (8)  2!)()r)(;24        (11)  7.0633 

(3)  1416.5     (6)  .200375     (9)  .00001328     (12)  1202800 

(13)  11^,  carry  the  decimal  to  3  places 

(14)  H         (15)  5^. 

389.  Conversely :  To  Find  the  Significant  Figures  of  a 
Number  from  its  Mantissa. 

1.  When  the  given  mantissa  is  the  same  as  one  in  the 
table. 

The  first  two  figures  of  the  number  are  found  in  the  same 
row  at  the  left  in  the  column  marked  ''  N."  The  third  is 
at  the  top  of  the  column  in  which  the  given  mantissa  is 
found. 

Find  the  number  corresponding  to  1.3118,  its  logarithm. 
Turning  to  the  table,  the  mantissa  is  found  opposite  20 
and  5. 

Hence  20.5  is  the  number.     (388,  1.) 

2.  When  the  given  mantissa  lies  between  two  in  the  table. 
Here  the  number  is  interpolated  in  the  same  way  that  the 
logarithm  was  before. 


TOPICS    KKLATED   TO    Ei^t'ATlONS.  405 

Thus,  given  ,3.847«S  to  find  tlic  huiiiIkm-  coirespondinii:. 
R«'ferring  to  the  table,  we  find 

:i847()  (next  less)        =  log.  7()4(>  (See  table.) 

.S.H478  (given  log.)      =  log.  7043.33         (By  interpolation.) 
3.S482  (next  greater)  =  log.  7050  (See  table.) 

Tabular  difference  =  G ;  78  —  70  =  2 ;  (>  :  2  :  :  1  :  a;  .-.  .r 

=  2  -f-  0  =  .a-^i 

The  charaxiteristic  being  3,  there  are  4  integral  figures 
in  the  number  (388),  which  places  the  decimal  after  the 
first  3.     And  so  in  all  ca.ses. 

Hence  to  find  a  number  when  its  logarithm  is  given. 

(1.  From  the  given  mantissa  take  the  next  less  found 
in  the  table,  and  divide  the  remainder  by  the  tabular  differ- 
ence following  the  latter. 

(2.  The  resulting  decimal  figures  are  to  be  annexed  to 
the  three  figures  corresponding  to  the  lesser  mantissa. 

(3.  The  decimal  point  is  then  inserted  between  the 
figures  in  accordance  with  the  rules  of  the  j)receding  article. 

390.  Exercise  in  Finding  Numbers  from  their  Logarithms. 

1.  Find  for  pracvtice  the  number  corresponding  to  3.201(). 
In  the  table  we  find  the  next  less  mantissa  to  be  .2014,  cor- 
responding to  159,  and  the  tabular  difference  27.  Subtract- 
ing and  dividing,  2  -j-  27  =  .074  -f-  •  Annexing  these  figures 
to  151),  we  have  159074  -|-  .  Now.  tlie  given  characteristic 
is  .'J.  Hence  4  figures  prectuU'  tlic  decimal  ]H)int.  ;nid  fho 
numl)er  is  1590.74  +  . 

2.  4.8016       6.   2.0095      8.   4.7320       11.   9.8423-10 

3.  2.1144        6.    4.248S       9.    1.0410       12.    7.0453-10 

4.  0.44HS        7.    1.94SS     10.    3.0210       13.    ().52()9  —  10 

391.  Uses  of  Systems  of  Logarithms.  —  Logarithms  wcic 
invented  to  abridge  the  lalKu- of  multiidication  and  division. 
They  are  used  to   multii»ly.  divide,  raise   to  ]K>wtMs.  and 


406  TEXT-BOOK  OF  ALGEBRA. 

extract   roots,  as  well   as  in   the   solution  of   exponential 
equations. 

392.    To  Multiply  Numbers  by  the  Use  of  Logarithms. 

1.  Let  us  lind  tlie  product,  for  example,  of  155,  207 
and  939. 

155  =  W'-'^''  (See  table,  log.  155) 

207  =  102-3i«>  (See  table,  log.  207) 

939  =  10-^-«'-^^  (See  table,  log.  939) 

.-.  155  X  297  X  939  =  10' -^^^^  (Ax.  3) 

(Adding  the  exponents  of  10  by  the  rule  for  the  multipli- 
cation of  monomials.) 

By  Definition,  7.4790  is  the  logarithm  of  the  product 
sought.  Referring  to  the  table  to  find  the  number  corre- 
sponding, it  is  found  to  be  30128571  +.  Hence  155  X  207 
X  939  =  30128571,  approximately. 

(=  30127815  exactly.) 

Thus  multiplication  is  made  to  depend  on  addition,  a 
process  much  more  easily  performed. 

2.  Hence,  to  multiply  together  two. or  more  numbers, 
first  find  their  logarithms  and  add  them,  and  then  find  the 
number  corresponding  to  this  logarithm. 

3.  Multiply  together  the  following  numbers  by  loga- 
rithms, obtaining  the  corresponding  approximate  results. 

(1.    Multiply  29  by  39. 
log.  29  =  1.4624  (Table,  log.  from  number.) 

log.  39  =  1.5911  (Table,  log.  from  number.) 

log.  1131  4-      3.0535       (Table,  number  from  logarithm.) 

.-.  29  X  39  =  1131  +  approximately  (=  1131  exactly). 

(2.    19  X  37.  (4.   85  X  10  X  21. 

(3.    43  X  68.  (5.    122  X  133  X  144. 

(6.12  X  13  X  14  X  15  X  16  X  17. 


TOPICS    ItKLATF.n    TO    Kgi'ATlONS.  407 

393.    To  Divide  Numbers  by  the  Use  of  Logarithms. 

1.  o  hiid  the  quotient,  e.  g.,  of  UU7  divided  by  l.\)o. 
207  =  lO-2-8i«>  (Table,  log.  from  number.) 
1.93  =  loo-^^'^c  (Table,  log.  from  number.) 
...  207  -r-  1.93  =  10=^'«*^*  (Ax.  4) 

(Subtracting  the  exponents  of  10  by  the  rule  for  the 
division  of  monomials.) 

Now,  2.0304  is  the  logarithm  of  the  quotient.  Turning 
to  the  table,  we  find  for  the  number  corresponding,  107.25. 
Thus,  division  is  made  to  depend  upon  subtraction. 

2.  Hence,  to  divide  one  number  by  another,  first  find 
their  logaritlims  and  subtract  the  logarithm  of  the  divisor 
from  that  of  the  dividend,  and  then  find  the  number  corre- 
sponding to  the  difference  of  their  logarithms. 

3.  Perform  the  following  divisions  by  logarithms : 

(1.   Divide  35  by  7. 

log.  35  =  1.5441  (Table,  log.  from  number.) 

log.    7  =  0.8451  (Table,  log.  from  number.) 

log.    5  =    .6990  (Table,  number  from  log.) 

(2.   91  -^  13.  (5.   2345  -4-  163.5. 

(3.   85-^17.  (6.   796.325 -r- 196275. 

(4.   198 -^  2.67.  (7.   .00367^2.61. 
(8.   .01917 -4- .00021. 

4.  Rule  for  evaluating  Com|X)und  Expressions,  such  as 

9  X  13  X  103  ,  .  ,        .^, 

— — — —  by  means  of  logarithms. 

to  X  87 

(1.  Find  in  turn  the  logarithms  of  the  numerator  and 
denominator  by  adding  the  logarithms  of  their  respective 
factors. 

(2.  Subtract  the  latter  sum  from  the  former,  and  the 
number  corresponding  is  the  quotient  sought. 


408  TEXT-BOOK    OF    ALGEBUA. 


Find  tlie  value  of 

(1.     '^^J^^J^-^.     Model  Solution. 
^  73  X  87 

log.      9  =  0.9542  log.  7:3  =  1.8633. 

log.    13  =  1.1139  log.  87  =  1.9395 

log.   103  =  2.0128  denominator  =  3.8028 
numerator        4.0809 

denominator    3.8028  .      9   X    13    X    103  _  ^  ^^^ 

1.897     0.2781  *  *       73  X  87              '  '    * 

^2     336.8  X  37  .^     212  X  6.13  X  2009 

^  '    7984  X  22  •  ^  '    365  X  5.31  X  2.576  ' 

.o             .07654  .^       .0062  X  .0007  X  2 


(6. 


83.947  X  0.8395  '       3.6  X  .00005  X  9.764 

(2^-  X  61)  -^  U 


(271-206)81 
(7.    Find  x  in  28.035  :  3.278  =  3114.27  :  x. 

394.    To  Raise  Numbers  to  Powers  by  the  Use  of  Logarithms. 

1.  liaise  13  to  the  liftli  power. 

13  =  10i-"2» 

135==105-^««^  (Ax.  5) 

(Multiplying  the  exponent  of  10  by  5  in  accordance  with 
the  rule  for  raising  to  powers.) 

Finding  the  number  corresponding  to  the  logarithm 
5.5695,  we  have  13^  =  371090.9,  approximately. 

2.  To  raise  a  number  to  a  i)ower  multiply  its  logarithm  by 
the  index  of  the  power  and  find  the  number  corresponding. 

3.  Perform  the  operations  indicated  in  the  following 
exercise. 

(1.    Kaise  6  to  the  tliird  power. 

log.  6  =  0.7782  (Table,  log.  from  number.) 

log.  216  +  =  2.3346  (Table,  number  from  log.) 

.'.  6^  =  216  +,  approxim.ately. 


TOPICS  i:klati:i)  to  hquations.  409 


(2.  :*=? 

(7.  (TA)*. 

(3.   5a*. 

(8.    (.9975)". 

(4.    (W- 

(9.    (33.9  X  43.4  -r-  3814)". 

((>.    (l..%72)". 

(10.  (W)'. 
..    /. 000106 y 

■     ■    \'My  X  .07/  * 

395.  To  Extract  the  Roots  of  Numbers  by  Means  of  Loga- 
rithms. 

1.  Since  the  laws  of  exponents  hold  for  fractional  as 
well  as  integral  i)owers,  the  rule  of  the  preceding  article 
holds  good  here :  hence,  multiply  the  logarithm  of  the 
given  number  by  the  fractional  exponent  of  the  power  to 
which  the  numl>er  is  to  be  rais<Ml. 

If  a  root  is  to  be  extracted,  say  the  fifth,  the  logarithm 
would  l)e  multiplied  by  ^  (or  divided  by  5)  as  in  the  extrac- 
tion of  roots  of  monomials. 

2.  EXKRCISK. 

(1.    (35()21)l 

log.  35021  =  4.5444 

3  X  4.5444  =  2.720  - 

iog.  533.  =  2.7267  .-.  (35021)1  =  533. 
(2.    (.0069)i  log.  .0069  =  3.8388. 

The  log.  3.8388  is  now  to  h*  divided  by  5.  If  the  division 
were  made  ns  the  lognrithm  sttunls,  the  result  would  be  the 
same  lus  when  tlie  characteristic  is  pins,  wliich  is  manifestly 
wnmg.  The  expression,  as  was  shown,  is  really  a  binomial, 
and  must  be  divided  in  the  binomial  form.  This  is  accom- 
plished by  hori'owhn/.  so  as  to  make  the  nffjatwe  ehararf eris- 
tic divlsiblr. 

],,.4.  (.(MK;<))i  =  J  (718388) 

=  i  (^>  + -'.«'^^'^)    _ 
log.  .3697  Ans.  =  1  -(-  .."')(;78  =  1.5678. 


410 


TEXT-BOOK    OF    ALCJEBRA. 


(3.    8^ 

(7.    11^ 


(4. 
(8. 


(11 


^  V(f) 


/^49         ,,^_    ^73567 
V  1991        ^ 

(12.   ^/m  X  V117     (13. 


(6.    906.80* 
(10.   2.5673" 

16^  X  15^  X  14* 
13^  X  12i  X  11 


29.62^ 
396.    Accuracy  of  Results  obtained  by  using  Logarithms.  — 

A  little  consideration  will  show  that  in  getting  a  number 
from  its  logarithm  (4-place  table)  one  cannot  be  sure  of 
more  than  3  figures.  As  a  general  thing  the  fourth  and 
often  the  fifth  will  be  correct,  but  the  latter  is  very  un- 
certain. 

Besides,  the  logarithm  Itself  may  be  somewhat  in  error  from  the 
circumstances  under  which  it  was  obtained. 
Thus,  dividing  771  by  119 


FOUIt-PLACE. 

log.  771  =  2.8871 
log.  119  =  2.0755 


SIX-PLACE. 

2.887054 
2.075547 


log.  6.480  =  0.8116 


log.  6.479  =  0.811.507 


Here  four-place  logs,  are  in  error  by  about  a  half  unit  of  the  fourth 
place,  as  six-place  values  show.  These  errors  are  superadded  in  the 
quotient  logarithm,  and,  as  a  result,  the  four-place  table  gives  6.480, 
while  the  more  accurate  six-place  table  gives  6.479. 

In  raising  to  powers  an  error  may  be  considerably  increased.  Thus, 
(3g2)t  worked  out  by  a  four-place  table  gives  rise  to  an  error  of  about 
4  units  in  the  fourth  place,  the  reason  being  that  the  log  of  32  is  too 
small,  and  that  of  6  too  large  in  the  table,  and  the  error  doubled  by 
subtraction  is  then  multiplied  by  4. 

We  subjoin  a  little  table  of  errors 


PUOBLEM 

ANSWEK  BY  LOGAHITHMS 

TliUE    VALUE. 

155  X  207  X  939 

207  ^  1.93 

9  X  13  X  103 

71  X  75 

290^ 

301285714- 
107.25 

2.2621-h 

2.5729 

30127815 
107.254 

2.2631 

2.5728  nearly. 

TOPICS   i;i:i..\  ri;i)    !'( >   i:(.)rA'ri()Ns.  411 

Tlie  student  is  strongly  urged  to  verify  by  actual  nuiltiplications 
and  divisions  all  problems  which  can  be  so  solved  without  too  great 
an  expenditure  of  time  and  labor.  A  fair  idea  of  the  accuracy  of 
logarithmic  computations,  and  the  advantage  derived  in  their  use 
may  thus  be  gained.  If  other  (six-  or  seven-place)  tables  are  at  hand 
let  them  be  tested  in  the  same  way.  Every  endeavor  should  be  made 
to  get  a  practical  as  well  as  theoretical  undei-standing  of  the  sub- 
ject at  the  start.  Constant  use  is  made  of  logarithms  in  trigonometry 
and  other  branches  of  mathematics. 

The  student  might  be  led  to  suppose  that  if  logarithms  give  results 
correct  only  to  a  certain  number  of  places  (depending  on  the  number 
of  decimal  places  in  the  table  used),  not  much  practical  use  could  be 
made  of  them.  But  such  is  not  the  case.  In  actual  measurements, 
as  with  a  rule,  it  is  not  possible  to  get  lengths  any  closer  than  hun- 
dredths of  an  inch,  and  often  not  so  close.  Hence  results  obtained 
by  logarithms  may  be  made  correct  within  the  limits  of  error  by 
choosing  a  table  having  a  sufficient  number  of  places. 

397.  *  Logarithms  to  other  Bases  can  be  derived  from  the 
Briggsian  Logarithms.  —  To  show  this  the  following  theorem 
must  be  proved  :  — 

Theorem  :  If  the  logarithm  of  any  number  he  taken  to 
two  different  bases,  thejirst  logarithm  equals  the  second  mul- 
tijdied  by  the  logarithm  of  the  second  base  in  thejirst  system. 

Let  n  be  any  number  and  b  and  b'  two  bases. 
Put  log.^ /I  =  jc, and  log.^ b'  ^y\  then  by  definition, b'  =  /^'^ 
and  n  =  V  =  {Jyyy  =  ^^, 

i.e.,  log.ft  n  =  xy  =  log.ft  n  X  h)g.,,  //'     n.  h\  />. 

Thus,  log.,0  5  =  log.2  5  X  log.,,,  L'. 

Referring  to  the  values  as  given  in  383.  wo  have 

.69*)0  =  2.3223  X.3010 

which  is  easily  verified  by  multiplication. 

If  log.io  n  =  log.^.  n  X  log.,0  b' 

log.^  n  =  log.,0  V  4-  log.,0  b'  (Ax.  4) 


412  TEXT-BOOK   OF    ALGEBRA. 

Hence,  to  find  the  logarithm  of  any  niimhev  to  a  new  base, 
divide  the  member'' s  lofjarithm  to  base  10  by  the  logarithm  of 
the  new  base  to  base  10. 

The  student  can  easily  verify  that  this  is  true  in  the 
examples  of  383. 

398.  Solution  of  Exponential  Equations  by  Means  of  Loga- 
rithms. 

1.  To  solve  the  equation  a^  =  b. 

Since  the  two  members  of  the  equation  are  equal,  their 
logarithms  are  equal  (general  axiom),  and  since  log.  (a^)  = 
X  log.  a  (395),  we  have 

X  log.io  a  =  log.io  b 

.^_log^  (Ax.  4) 

log.io   ^ 

Remark.  —  This  result  may  also  be  derived  by  means  of  the 
theorem  at  the  end  of  the  last  article,  x,  b,  and  a  equaling  respec- 
tively, log.yii,  n,  h\ 

2.  Exercise. 

(1.    Solve  the  equation  13^  =  129. 

log.io  129      2.1106      ,  „,,,      , 

X  =    /  ^"    ,  .,    =  zr-pr^  =  1.895       A71S. 

log.io  l':5       1.1139 
(2.   2^  =  64.  (6.   9.1^=2467.2 

(3.   2(K  =  100.  (7.   3'^''  +  i  =  27. 

(4.    IP  =  3.  (8.    .052^'- ^1  =  396.2 

(5.   3-- =.8  (386,  a)  (9.    lO^-^ +  2 X  10-^  =  80. 

Suggestion.  —  Solve  first  as  a  quadratic  for  10*. 

(10.    Given  a'^^b"^  =  e 

Taking  logarithms  of  both  sides 

log.  «"'^  H-  log.  b"^'  =  log  c  (392) 

or,  mx  log.  a  +  nx  log.  b  =  log.  c  (394) 

log.  c  . .       ,x 

.-.    X  =  ^i -^ — 1 y  (Ax.  4) 

m  log.  a  -f  n  log.  b  ^  -^ 


TOPICS    IIKLATKIJ    TO    K(M  ATIONS.  413 

(11.   L'^  ;>i^  =  LMMM). 

Suo«KsTiox.  —  Put  r-  —  4x  4-  5  .==r  z  Jind  solve  for  2. 


414  TEXT-BOOK  OF  ALGEBRA. 


CHAPTER  XXVII. 

ARITHMETICAL    AND    GEOMETRICAL    PROGRESSIONS. 

399.  An  Arithmetical  Progression  is  a  series  of  terms 
which  increase  or  decrease  by  a  common  difference. 

E.g.,  in  4,  7,  10,  13,  16,  etc.,  the  common  difference  is  3 ; 
and  in  8,  6,  4,  2,  0,  —  2,  —  4,  etc.,  the  common  difference  is  2. 

In  the  study  of  progressions  there  are  two  problems 
which  stand  out  prominently;  viz.,  to  find  the  nth  term,  and 
to  find  the  sum  of  n  terms. 

400.  Formula  for  finding  the  7ith  Term  of  an  Arithmetical 
Progression. 

Let  a  be  the  first  term,  d  the  common  difference,  and  n 
the  number  of  terms  considered.     Then,  by  definition, 

«,  a  -j^  d,  a  ^2d,  a  ^'dd,  a  ^4:d,  etc., 
is  the  arithmetical  progression,  the  signs  being  taken  all 
positive  or  all  negative  according  as  it  is  an  increasing  or 
decreasing  series  ;  for  each  term  is  greater  or  less  by  d  than 
the  one  which  precedes  it. 

We  are  to  find  the  nth  term. 

Let  us  write  several  terms  with  the  ordinal  of  each 
above  it. 

a,       a  +  d,       <i  +  2  d,      a  +  3  r/,    .     .     a  -\-  (n  —  1)  d. 
a,       a  —  r/,       <i  —  2  c/,       (L  —  3  d,    .     .     a  —  (n  —  1)  d. 

It  will  be  seen  that  the  multiplier  of  a  is  always  1  less 
than  the  number  of  the  term.  Hence,  the  nth  term  is  of 
the  form  given  above. 


•I'Ml'MS     I;i:i.A  Ii:i>     I*  )    l.(M    AlIoNS.  415 

Calling  /  the  /itli  or  last  term, 

(1)    ^  =  a  _1_  (n  -  1)  d. 

Hence,  to  find  the  last  term,  multiply  the  common  diifer- 
ence  by  one  less  than  the  number  of  terms,  and  add  the 
product  to  the  first  term  if  an  increasing,  or  subtract  from 
the  first  term  if  a  decreasing  progression. 

401.  Formula  for  finding  the  Sum  of  //  Terms  of  an  Arith- 
metical Progression. 

As  lu'fore,  call  a  the  first  term,  d  the  common  difference, 
?i  the  number  of  terms,  and  /  the  last  term.  Denote  by  &• 
the  sum  of  7i  terms.  If  rf  be  added  each  time,  we  may 
write  the  last  term  /,  the  next  to  the  last  I  —  c?,  the  second 
from  the  last  I  —  2d,  etc.  But  if  d  is  subtracted  each  time, 
the  next  to  the  last  term  would  be  d  (jreater  tlian  /,  or  I  -f-  d, 
the  second  term  from  the  last  /  +  1' '/,  etc.  (See  45,  i,  and 
remark  on  =p  Art.  344.)     Consequently  we  may  write 

a,il^^d,a^^2d,a^^^d,    .    .    l^'Sd,l^2d,l^d,L 

Trying  to  sum  this  series,  a  little  study  suggests  the  plan 
of  adding  the  first  and  last  terms,  the  second  and  next  to 
the  last  terms,  the  third  and  third  from  the  last  terms,  and 
so  on,  since  the  sum  in  each  case  would  be  the  same,  viz., 
a  +  /.  It  will  l>e  a  little  plainer,  however,  to  add  the  series 
to  itself,  writing  the  seccmd  in  reverse  order,  as  this  also 
will  give  the  sum  (f  -\-  I  for  each  addition,  and  it  is  clear 
that  tliere  will  l>e  just  n  sudi  additions. 

N  =  (/-}-«  i  </  -f-  tf  Jt:  -  (^  +  ('  ±  •'  "^  +  •    •  ^ 

jy  =  /-t-/=P  rf  -h  /  =f  2  r/  -f  /  =f  3  d  -\-  .    .    .  a 

28  =  a  ^r^7i~^T'^a7-f7  -{-  a  -f-  / -}-    .  a  -f-/(Ax.  1) 

2  s  =  a  -\-  I  counted  n  times  since  there  are  //  terms. 

(2)  .V  =  |;  (<,  4-  /). 

Hence,  to  find  tin*  sum  of  an  arithmetical  progression, 
after  calculating  the  value  of  the  hust  term  add  it  to  the 
first  term  and  multiply  thr  >uin  l»y  lialf  the  number  of 
terms. 


416  TEXT-BOOK  OF   ALGEBRA. 

402.    Other  Problems   in  Arithmetical    Progression.  —  The 

progressions  furnish  excellent  exercise  in  the  solution  of 
such  equations  as  were  given  in  44-50  of  Art.  216.  Thus, 
in  equation  (1)  above,  Z  =  a  -j-  (71  —  1)  d,  there  are  four 
quantities  involved  (s  not  appearing),  three  of  which  being 
given,  the  fourth  may  be  calculated.  This  gives  four  prob 
lems.  Then,  similarly,  equation  (2)  gives  four  more  involv- 
ing a,  n,  I,  and  s,  (the  fifth  quantity,  d,  not  appearing). 
Next,  by  eliminating  n  between  (1)  and  (2),  we  have  an 
equation  involving  a,  d,  I,  and  s.  Eliminating  a  between 
(1)  and  (2),  the  resulting  equation  contains  d,  n,  I,  and  s. 
Last  of  all,  eliminating  /,  the  resulting  equation  contains  a, 
d,  71,  and  s.  Consequently  there  are  five  equations,  each 
one  of  which  gives  four  problems. 

As  an  example,  suppose  /,  d,  and  s  given  to  find  a.  Since 
?«-  is  the  omitted  quantity,  and  (1)  and  (2)  both  contain  it, 
it  must  be  eliminated.     From  (2) 

^1  =  "-  (^^-^^ 

Substituting  this  value  in  (1) 

J  ,    (2  s  —  a  —  I)  d 

l  =  a  +  ^ ,    ,    ^ 

a  -\-  I 

in  which  a  is  supposed  to  be  the  unknown. 

al  +  P  =a'^-^al-{-2  ds  -  ad  -  Id  (Ax.  3) 

a^  -  da  =  l^^ld-2  ds  (Ax.  a) 

4  a''  -^da-{-d'  =  4:P  +  4.ld-i-  d'  -  S  ds  (335) 
2a  —  d=J^  V(2  l-^dy-  <S  ds. 

'^  =  r>  i  9  V(2  l^dy-Sds     Atis. 

1.  To  find  I,  being  given  (1.  a,  d,  71  (2.  a,  71,  s  (3.  a,  d,  s 
(4.  d,  n,  s. 


TOI'KS     KKLAIKl)     In     I  .(,H   A  T  l<  iN  S.  417 

2.  To  find  .v.  Ijeiiig  given  (1.  '/.  /,  //  i'J.  c  I.  d  (.'1  a,  d,  n 
(I.   /.  <l.  „. 

3.  To  find  </.  Ikmu^^  ^nvm  i  1.  n ,  //.  /  (L\  a.  n,  s  ('.\.  a,  /,  .s 
(4.   /.  //.  .V. 

4.  To  tiiul  //.  l)eiiig  given  (1.  o.  d.  I  CI.  a.  <l.  s  (.>.  o,  /,  n 
(4.  /.  f/.  .V. 

5.  To  find  <i,  l>eillg  given  (1.   d.  n,  I   (1*.   d,  //,  .s   (.'>  /,  //.  n. 

XoxE.  —  Of  these  problems  only  one  is  ooninionly  given  special 
coiisideration.  It  is  that  in  which  a,  /,  and  n  are  given  to  find  d. 
It  may  be  stated  in  MOrtls  as  follows:  required  to  insert  it  —  2  arith- 
metical means  («  and  /  being  the  other  two  terms)  between  any 
two  quantities  n  and  /.  After  having  found  the  common  difference, 
it  is  an  easy  matter  to  write  down  tin  vc  mtans.     Tin*  formula  for  f/, 

/       ,/ 
as  given  among  the  answers,  is  </ 

If  7  means  are  to  be  inserted,  //       '•.  ami  so  ni  general. 

403.    Exercise  in  Arithmetical  Progression. 

1.  Find  the  last  term  and  the  sum  of  the  terms  in  the 
series  5,  7,  9,  etc.,  to  20  terms. 

Here  rt  =  5,  rf  =  L>,  7t  =  20. 

To  find  /,  /  =  a  +  (7?  -  1)  rf  =  5  -f  19  X  2  =  43.   Am. 

To  find  N,  .^  =  I  {a  4-  /)  ^  10  (5  -h  43)  =  480.    Ans. 

2.  Find  the  17th  term  in  the  series  10.  11.^,  13,  etc. 

3.  Find  the  last  term  and  the  sum  m  ."»,  9,  13,  etc..  to  19 

4.  Find  the  L'tHh  terni  of  the  series  T),  1,  —  3,  etc. 

5.  Find  the  wth  term  of  2,  2^,  2§,  etc. 

6.  Sum  7  -h  V  -|-  V'  +  ^*<'v  to  16  terms. 

7.  Find  the  last  term  and  the  sum  in  —  ].  —  ;].  etc.  to 
iti)  terms. 

8.  Sum  2  a  —  5  h,  7  tf  —  'J  b,  etc.,  to  9  terms. 

9.  (liven  ^/  =  5,  c/  =4,  and  I  =  201,  required  /i,  and  /f. 
10.    Insert  12  means  between  12  ;tn<l  77. 


418  TEXT-BOOK    OF    ALGEBRA. 

11.  Insert  9  means  between  2  and  5  and  write  the  pro- 
gression to  8. 

12.  Given  a  =  —  B,  I  ==  —  47,  s  =  —  1118,  find  7i  and 
then  d. 

13.  What  is  the  arithmetical  mean  between  4  and  10? 
between  16  and  —  4  ?  between  a^  -{-  ab  —b'^  and  a^  —  ab  -\- 

14.  The  first  term  of  a  series  is  2  and  the  common  differ- 
ence I.     What  term  will  be  10  ? 

15.  Which  term  of  the  series  5,  8,  11,  etc.,  is  320  ?      . 

16.  Find  the  sum  of  —  'Sq,  —  y,  q,  etc.,  to  p  terms. 

17.  How  many  terms  of  the  series  .034,  .0344,  .0348,  etc., 
amount  to  2.748  ? 

18.  Given  d  =  —  ?,,  I  =  —  39,  s  =  —  264  to  find  a  and  ?^. 

19.  How  many  terms  of  the  series  5  -f  7  -|-  9  -}-  etc., 
must  be  taken  in  order  that  the  sum  may  be  480  ? 

Remark.  —  Negative  values  of  n  are  by  the  nature  of  the  subject 
excluded. 

20.  Show  that  if  it  is  known  that  the  7}ith  term  of  a 
series  is  p,  and  the  nth  term  is  q,  that  the  series  is  known. 
Find  d,  a,  and  I. 

21.  Find  the  series  in  which  the  27th  term  is  186,  and 
the  45th  term  222. 

22.  The  sum  of  15  terms  of  an  arithmetical  progression 
is  600,  and  the  common  difference  is  5.     Find  the  first  term. 

23.  Of   two   arithmetical    series   whose   first   terms   are 

equal,  the  first  has  for  its  last  term  39,  and  for  sum  207 ; 

while  the  second  has  for  its  last  term  124,  and  for  its  sum 

917.     What  is  the  first  term  in  both,  and  what  the  number 

of  terms  in  each  ? 

2.S 
►SuGO?:sTioN.  —  See  Art.  250.     Use  formula  n  =^-f-, — • 

I  +  a 


T(>1M(  S    Ll.l.A  ri:i)    i«>    iAji  ATIONS.  410 

24.  Two  huiidied  stones  beiug  placed  on  tlie  ground  in  a 
straight  line  at  the  distance  of  *2  feet  from  each  other,  how 
far  will  a  person  travel  who  shall  bring  them  separately  to 
a  basket  whicli  is  placed  20  yards  from  the  first  stone,  if  he 
start  from  the  spot  where  the  basket  stands  ? 

26.  A  body  falls  l(),».j  ft.  the  iii-st  second,  and  in  eacli 
succeeding  second  32^  ft.  more  than  in  the  next  i)receding 
one.  How  far  will  it  fall  during  the  Kith  second,  and  what 
will  be  the  whole  distance  fallen  through  in  16  seconds  ? 

26.  If  a  person  saves  $1(>(>  and  puts  it  at  simple  interest 
at  r}f/c  at  the  end  of  each  year,  how  much  will  his  property 
jimount  to  at  the  end  of  20  years  ? 

27.  A  man  wius  paid  for  drilling  an  artificial  well  3.24 
marks  for  the  first  meter,  3.29  marks  for  the  second,  3.34 
marks  for  the  third,  and  so  on.  The  well  had  to  be  sunk 
500  meters.  How  much  was  paid  lor  the  last  meter,  and 
how  nmch  for  the  whole  ? 

28.  A  travels  uniformly  20  miles  a  day ;  B  starts  3  days 
later  and  travels  8  miles  the  first  day,  12  the  second,  and  so 
on  in  arithmetical  jjrogression.  In  how  many  days  will  H 
overtake  A  ? 

29.  Show  that  if  the  same  quantity  be  added  to  every 
term  of  an  arithmetical  i)r()gressi()n  the  sums  will  be 
in  a.  p} 

30.  Show  that  if  every  term  of  an  a.  p.  be  multiplied  by 
the  same  (quantity,  the  products  will  be  m  a.  p. 

31.  The  sum  of  three  numl)er8  in  a.  p.  is  12,  and  the  sum 
of  their  scjuares  is  06 ;  find  them. 

32.  Find  four  integers  in  a.  p,  such  that  their  sum  sliall 
be  24,  and  their  product  1)45. 

33.  The  sum  of  five  numbers  in  a.  p.  is  45,  and  the  pro- 
duct of  the  first  and  fifth  is  five-eighths  of  the  product  of 
the  second  and  fourth.     Find  the  numl)ers. 

>  For  brevity  n.p.  will  be  used  for  aritliineticul  progression  and  g.  p.  for 
geometrical  progression. 


420  TEXT-BOOK    OF   ALGEBRA. 

404.  A  Geometrical  Progression  is  a  series  in  which  the 
terms  increase  or  decrease  by  a  common  ratio. 

E.g.,  3,  6,  12,  24,  48,  etc.,  the  common  ratio  being  2; 
and,  5,  1§,  |,  /y,  ^\,  etc.,  the  common  ratio  being  ^. 

405.  Formula  for  finding  the  7ith  term  of  a  Geometrical  Pro- 
gression. 

Let  a  be  the  first  term,  ?'  the  ratio,  w  the  number  of 
terms,  and  I  the  last  term.     Then,  by  definition,     - 

1st        2^        3^        4*^        5^^        6^h,  etc. 
a,        ar,       ar^,       ar^,       ar*,        ar^,  etc. 

Here  it  is  evident  that  the  exponent  of  r  in  any  term  is 
always  one  less  than  the  number  of  that  term.      Conse- 
quently the  exponent  of  r  m  the  nth  term  would  be  7i  —  1. 
Hence,  we  have, 
(1)    i  =  ar''-\ 

Or,  to  find  the  nth  term  of  a  geometrical  progression, 
raise  the  ratio  to  a  power  whose  exponent  is  one  less  than 
the  number  of  the  term,  and  multiply  the  result  by  the 
first  term. 

406.  Formula  for  finding  the  sum  of  7i  terms  of  a  Geometrical 
Progression. 

We  use  the  same  letters,  and  as  in  a.  p.  call  s  the  sum  of 
71  terms.  To  sum  the  geometrical  series  an  artifice  is  em- 
ployed analogous  to  that  used  for  summing  an  a.  j)-  The 
value  of  s  is  first  written  down,  and  then,  underneath,  the 
same  equation  multiplied  through  by  the  ratio.  By  sub- 
tracting the  first  equation  from  the  second,  all  the  inter- 
mediate terms  on  the  right  go  out,  leaving  only  the  first  and 
last  terms. 

s  =  a  -\-  ar  -\-  ar^  -f  a7^  -f-  ar^  -f- -f"  o^^"*"^ 

rs  =  ar  -\-  ar^  -\-  ar'  -\-  ar^  -f-  ar^  -\-    .   -j-  «r"~^  -|-  a.r^ 

rs  —  s       =    ar''  —  a     (Ax.  2)  [(Ax.  ?>) 


Tol'M 


l;i:i,ATl.l)    TO    KgLATlUNS.  421 


(r  -  1)  s  =  a  (/•"  -  1) 

Or,  to  find  the  sum  <>t  //  itims  of  a  ^.  7>.,  multiply  the 
first  term  by  the  quotient  ol  the  ratio  raised  to  the  ?ith 
jx)\ver  less  one,  divided  by  the  ratio  less  one. 

A  formula  in  which  n  is  replaced  by  I  is  readily  derived 
from  (2)  and  is  convenient  for  reference. 

(3)  5=^^:^ 


By  this  formula,  to  iind  the  sum.  multiply  the  last  term 
by  the  ratio,  from  the  product  -iilitiact  the  first  term,  and 
divide  the  remainder  by  the  ratio  less  one. 

Remark.  —  When  the  ratio  is  less  than  1,  the  use  of  negatives 
can  be  avoided  by  changing  the  signs  of  both  numerators  and  denom- 
inators in  (2)  and  (8). 

™.                «  ( 1  —  r")          ,          a  —  rl 
Then,  s  =       ,  -  ,  and  .h  = 

1  — /•  1  —  r 

407.  Other  Problems  in  Geometrical  Progression.  —  The 
three  eipiations  found  in  the  last  two  articles  are  really 
equivalent  to  only  two  indej)endent  ones,  since  one  Avas 
<!•  iiv.'d  from  the  other  two.  I*r:i(  tieally  the  same  state- 
ment (ran  be  made  here  as  that  at  the  beginning  of  402,  d 
being  replaced  by  r.  The  actual  work  of  elimination  and 
solution  in  ff.  p.  is  far  more  difficult  to  accomplish  than  in 
a.  p.  Indeed,  in  four  cases  the  general  solution  of  the 
problem  cannot  be  obtained. 

1.  Problems  which  can  he  solved  without  the  use  of 
logarithms. 

(1,    Let  it  be  required  to  find  s  in  terms  of  r,  /,  and  «, 
(I  being  eliminated.     From  (1). 


422 


TEXT-BOOK    OF    ALG 

KBRA. 

I 

(Ax.  4) 

Ir  -  a          '        /'"-^ 

(228) 

'  -    r-1            r-1 

Zr'*  -  I 

,."-1              /  (r-  _  1) 

(182) 

'  -    r-1         r--^{r-l) 

{^) 


1 

(2.  To  find  /,  being  given  (1)  a,  r,  s,  (2)  r,  n,  s. 

(3.  To  find  a,  being  given  (1)  r,  n,  I,  (2)  r,  n,  s,  (3)  ?%  /,  s. 

(4.  To  find  r,  being  given  (1)^  a,  n,  I,  (2)  a,  I,  s. 

(5.  To  find  s,  being  given  a,  7i,  I. 

2.    Problems  which  can  be  solved  by  the  use  of  logarithms. 

(lo    Given  a,  I,  and  s,  to  find  n. 
Here  r  must  be  eliminated.     Solving  from  (3). 

rs  —  s  =  Ir  —  a  (Ax.  3) 

rs  —  Ir  =  s  —  a 
s  —  a 

'■  =  .— y 

Substituting  this  value  in  (1)  _  =  r""^ 

a 

I  _  ( s  —  a 

a  ~  \V-^ 

Solving  this  equation  for  7i,  as  in  398. 

^  —  1  =  log.  10  (  -  )  -r-  log.  10 


n  =  log.io  I  —  log.io  a_ ^ 

log.io  {s  —  a)  -  log.io  {s  —  l)^    ' 

(2)  To  find  n,  being  given  (1)  a,  r,  I,  (2)  a,  r,  s,  (3)  r,  /,  s. 

1  To  insert  »  — 2  geometrical  means  between  a  and  /,  r=    "\/— •    I-'ftm  be  the 
number  of  means  to  be  inserted.    Then  r  =  "'\/ '  . 


T<UM(  >     Ul.l.A  lKi»     1<»     i.(.>r  A  IK  »NS.  4'28 

3.  l*r()l)leins  which  lend  to  the  solution  of  equations 
higher  than  the  second. 

(1)  For  example,  given  o.  ,i.  s.  to  tiiul  /.     Here  r  must 

be  t'liniinatcd.     As  shown  above,  r  =  ^  _  . .     Substituting 
this  value  in 

(1),  /  =  a  ^^-^^:y"\  or  /  (s  _  /)»-!  =  </  (N  -  a)"    \ 

which  is  an  equation  of  the  nth  degree  in  /. 

(2)  Given  n,  /,  «,  to  find  a. 

(3)  To  find  r,  being  given  (1)  a,  ti,  s,  (2)  «,  /,  3. 

408.  Summation  of  a  Decreasing  Geometrical  Progression 
which  extends  to  Infinity. 

In  a  geometrical  progression  m  which  the  ratio  is  less 
than  unity,  the  terms  grow  smaller  and  smaller  the  farther 
the  series  is  carried,  and  approach  the  limit  zero.  (See 
358,  1.)  At  the  limit,  therefore,  when  an  infinite  number 
of  terms  is  taken,  the  last  term  equals  zero.  Putting  1  =  0 
.in  the  value  of  s  given  in  (3) 

a  —  r/       a  —  r  X  0  a 

•'  =  1  -  r  ""      1-r      =  r^' 
or,  the  sum  of  a  decreiising  geometrical  series  which  extends 
to  infinity  equals  the  first  term  divided  by  one  less  the  ratio. 

As  an  example,  sum  the  series  2,  j|,  §,  5^,  .  .  .in  which 
the  tirst  term  is  2  and  the  ratio  is  \. 
2  ..       . 


s  = 


1-A 


409.  Repeating  Decimals  as  Examples  of  a  Decreasing  Geo- 
metrical Progression. 

A  repeating  decimal  is  one  which  repeats  certain  sets  of 
figures,  as  in  .92282828  .  .  .  the  two  figures,  28,  are  repeated 
to  infinity.     (See  290.  o,) 


424  TEXT-BOOK  OF  ALGEBRA. 

Writing  the  repeated   orders   as    common  fractions,  the 
first  term  and  ratio  are  readily  recognized  : 

i¥o  +  Tolj'o  0  +  TO  o¥o  0  0  +  etc. 
Here  the  ratio  is  yi^,  and  the  first  term  is  ^^^g. 


^100 

Hence  the  fraction  is  .92||  =  f^2_  +  ^||_  _  | ia§  _  ||||. 
In  the  same  way  any  repeating  decimal  can  be  reduced  to 
the  equivalent  common  fraction.  If  the  repeating  part  con- 
sists of  one  figure,  the  ratio  is  .1,  if  of  two  figures,  .01,  etc. 

410.  Exercise  in  Geometrical  Progression .  —  To  find  the 
ratio,  divide  any  term  by  that  preceding  it. 

1.  Find  the  ninth  term  and  sum  of  nine  terms  of  1,  3,  9, .  .  . 

2.  Find  the  ninth  term  and  the  sum  of  nine  terms  of  6, 
3,  11    .  .  . 

3.  Find  the  tenth  and  sixteenth  terms  of  the  series  256, 
128,  64,  ... 

4.  Find  the  last  term  and  sum  of  V2,  V6,  3  V2,  .  .  . 
to  12  terms. 

6.    Find  the  last  term  and  sum  of  8.1,  2.7,  .9, ...  to  7  terms. 

6.  Find  the  last  term  and  sum  of  —  f ,  ^,  —  ^, . . .  to  6  terms. 

7.  Given  r  =  10,  n=  9,  and  s  =  1,111,111,110,  required 
a  and  7i. 

8.  Find  the  sum  of  0.1  +  0.5  +  2.5  +  ...  to  7  terms. 

9.  Insert  6  geometric  means  between  ^i^  and  —  ,",.. 

10.  Insert  one  geometric  mean  between  a  and  J). 

11.  Show  that  of  two  unequal  positive  numbers  the 
arithmetic  mean  is  always  greater  than  the  geometric  mean. 

12.  Sum  to  infinity  the  series  9,  6,  4,  .    .  . 

13.  Sum  to  infinity  the  series   1  —  f  +  ott  —  rls?  •  •  • 

14.  Sum  to  infinitv  the  series  .9,  .0.*^,  .001.  .  .  . 


Topics   i;i:!.ati:i)  T(>  KgiATioNS.  425 

15.  Suiu  to  luHnity  the  series  .IV.V.V.^ 

16.  Sum  to  inrtnity  .37878;  also  .11^1351:^5 

17.  Show  that  the  reciprocals  of  the  terms  of  a  //.  p.  are 
also  III  fj.  p. 

18.  Find  the  Mim  ot  V«,  Vrt^  V«*,  etc.,  to  </  terms. 

19.  Find  tlie  series  in  which  tlie  lOth  term  is  320,  and  the 
(Jth,  -JO. 

20.  If  a  man,  whether  by  his  example  or  designedly, 
leads  a  single  fellow-man  from  the  path  of  rectitude  each 
year  during  twenty  years,  and  each  of  these  men  in  turn 
leads  astray  a  single  man  each  year  from  the  time  of  his 
change,  and  so  for  every  one  aft'ected,  what  will  be  the  total 
numl)er  led  astray  as  the  outcome  of  the  first  man's  had 
influence  ? 

21.  Achilles  pursued  a  tortoise,  which  was  one  stadium 
ahead  of  him,  with  a  speed  twelve  times  greater  than  that 
of  the  tortoise.  When  Achilles  reached  the  place  where  the 
tortoise  had  been  when  he  started,  the  tortoise  was  still  j*.^ 
stadium  in  advance  of  him ;  having  traversed  this  distance, 
the  tortoise  was  still  y^^  stadium  ahead  of  him,  and  so  on, 
ad  intinifnni.  Will  Achilles,  then,  never  catch  up  with  the 
tortoise  ? 

Explanation.  — This  problem  is  a  statement  of  Zeno's  celebrated 
sophism.  Let  x  equal  the  nuinl)er  of  stadia  which  Achilles  must  run 
to  <-atch  lip  with  the  tortoise.     Then  we  have  the  equation, 

X 

J"  —  rT=  1   .'.  •'■  =  li^i  stadia. 
.Mso.  summiuf;  the  progression  1  +■  yJj  +  ^H  "*"  n^ir  +  .  •  • 

Hence  the  sum  of  the  infinite  series  tends  to  a  definite  limit. 
Wliilo  it  is  perfectly  allowable  to  ronreite  of  the  distance  divided  up 
in  this  way,  the  sum  of  the  parts  is  only  li^r  stadia. 


426  TEXT-BOOK  OF  ALGEBRA. 


CHAPTER    XXVIII. 

INTEREST,  ANNUITIES,  AND   BONDS. 

411.  The  Problems  of  Interest  and  Annuities  require  the 
employment  of  the  formulae  of  progressions,  furnish  valu- 
able exercise  in  the  use  of  logarithms,  and  are  besides  of 
much  practical  importance. 

SECTION  I. 
Intekest. 

412.  Interest  is  a  payment  for  the  use  of  money.  The 
sum  lent  is  called  the  principal,  and  the  number  of  hun- 
dredths the  yearly  interest  is  of  the  principal  is  called  the 
rate. 

Three  kinds  of  interest  may  be  distinguished :  simple,  an- 
nual, and  compound. 

413.  Simple  Interest  is  interest  on  the  principal  alone.  It 
is  supposed  to  be  paid  annually.  But  when  it  is  not,  the 
total  amount  of  interest  (/)  due  at  the  end  of  n  years  is,  as 
was  stated  in  Ex.  47,  Art.  216,  i  =  prn. 

The  problems  of  simple  interest  may  be  summarized  as 
follows :  — 

1.  Given  j9,  r,  n,  to  find  /;  i  =  prn. 

2.  Given  p,  r,  i,  to  find  w,  n  =  i  -^  pr. 

3.  Given  p,  n,  i,  to  find  r  ;  r  =r  i  -^  pm. 

4.  Given  n,  r,  i,  to  find  p;  p  =  i  -^  rn. 

5.  If  ff  =  amount,  a  =^  p  ■\-  prn  =  p  {\  ~\-  rn). 


TOPICS    IIKLATKI)    TO    KQrATIONS.  42T 

From  this  formula  a  similar  set  ot  cases  may  be  distin- 
guished by  solving  in  turn  for  the  different  letters.  It  is 
not  necessary  to  go  into  this  further. 

414.  Annual  Interest  is  simple  interest  on  the  principal 
and  on  each  year's  interest  as  it  becomes  due. 

Let  p  be  the  principal,  r  the  rate  expressed  decimally, 
and  n  -\-f  the  time,  in  which  n  represents  the  whole  num- 
hev  of  years  the  principal  runs,  and  /  the  extra  fraction  of 
a  year. 

Now,  assuming  the  same  rate  for  the  unpaid  interest  as 
for  the  principal,  we  have  pr^  as  a  year's  interest  on  one 
payment,  pr.  But  the  first  payment  withheld  will  run  for 
//  -\-f—l  years  ;  the  second,  for  ?i  +  /  —  2  years  ;  the 
third,  for  n  -{■  f  —  S  years ;  and  so  on.  The  last  payment 
will  run  for  the  extra  fractional  part  of  a  year,  so  that 
altogether  there  will  be 

(''  +./•-  l)  +  (/i  +/-  2)  -f-  {n  -f-/-  8)  +  .  .  +  (1  +/)  +/ 
years  during  which  the  unit  payment  pr  will  draw  interest. 
Summing  this  arithmetical  progression  (401),  we  have 

Multiplying  a  single  year's  interest,  pr^y  by  this  total 
numl>er  of  years,  and  adding  the  product  to  the  sinij)le  in- 
terest, we  have  for  the  annual  interest  sought 

f,r  i  n  Jr.n  +  ^^^(/'  -h  lY-l  ^  =  pr  (  n  +/+  '^  (n  -f  !>/'_  1)  V 

KxAMTLK.  —  Calculate  thr  amount  of  $1.0(10,  for  1()  years 
inid  1?  months  at  annual  interest.  su))posing  the  rate  to  be 
Ans.  ^^lnL^ 

415.  In  Compound  Interest  all  unpaid  interest  is  added  to 
the  principal  a,s  fast  as  it  becomes  due. 

Call  p  the  principal,  a  the  amount.  ;•  the  rate  expressed 
decimally,  and  n  the  number  of  years.     We  may  have  the 

following  ]U"oblpms :  — 


428  TEXT-BOOK   OF    ALGEBRA. 

1.  To  find  the  amount  ivhen  the  principal,  rate,  and  time 
are  given. 

The  amount  of  the  principal  at  the  end  of  the  first  year 
is  p  (1  -\-  r).  Putting  this  amount  at  interest  during  the 
second  year,  the  amount  is  ^  ( 1  +  r)  (1  -j-  r)  =  j)  (1  -|-  r)^. 
Again,  putting  this  sum  at  interest  during  the  next  year, 
the  amount  becomes  p  (1  -^  ry,  and  so  on  through  the  n 
periods.     Hence, 

(1)  a=p(l  +  ry 

If  the  interest  be  compounded  q  times  a  year,  there  will 
be  q7i  periods  and  the  period  rate  will  be  -  and  the  formula 
becomes 

(2)  a^p{l  +  '^ 

If  i  is  the  interest,  then 

i  =p  (1  -\-ry—p  =p  [(1  +  ry  —  1]. 

Example.  —  Calculate  the  amount  of  a  note  for  $400, 
which  has  been  standing  for  8  years,  allowing  7  per  cent 
interest  to  be  paid  semi-annually. 

log.  a  =  log.  400  -f  log.  (1  +  •  Y-)''  (392,  2) 

=  log.  400  +  16  log.  1.035  (394,  2) 

=  2.6021  +  .2384  =  2.8405 

...  a  =  $692.67. 

Remark,  —  It  must  be  borne  in  mind  that  a  four-place  loga- 
rithmic table  gives  only  approximate  results.  Here  the  exact 
amount  is  $693.60.  In  this  chapter,  in  order  to  secure  the  proper 
degree  of  accuracy,  a  six-place  table  should  be  used. 

2.  To  find  the  present  worth  of  a  sum  payable  n  years 
hence,  allowing  compound  interest. 

Solving  from  (1)  above 

a  ^       .  1 

P  =  (1  ^  rY      Letting  j--py  =  v,  p  =  av\ 


TOPICS    KKi.AlLl*     i<>    Kl^HATIONS.  429 

XoxE.  —The  values  of  the  expression  (1  +  r)"  are  given  in  com- 
pound interest  tables  for  ditferent  values  of  the  arguments  r  and  )i. 
Using  these  lessens  the  labor  of  calculation.  The  logarithms  of  the 
c'.v  may  be  found  by  subtracting  those  of  (1  +  r)"  from  0. 

It  should  be  observ'ed  that  the  above  formula?  compound  the  in- 
terest in  the  last  fractional  period.  The  rules  commonly  given 
direct  to  calculate  simple  interest  for  the  odd  fraction  of  a  year. 
Compound  interest  in  this  case  favors  the  borrower. 

3.  To  find  the  time  ivhen  the  amount^  priiicipal,  and  rate 
are  given. 

Taking  logarithms  of  both  sides  of  a  —  j){\  -\-  /•)" 

log.  a  =  log.  7;  4-  log.  (1  +  ry  ( Ak.  8) 

log.  a  =  log.  p-\-n  log.  (1  4-  r) 

log.(l  +  r)    • 

ExAMi'LE. — Fin4  in  how  many  years  $1(M)  will  aniount 
to  $1050  at  T)  jxT  cent  eomiKmnd  interest.  Ans.,  4S.  17 
years. 

4.  To  find  the  rate  when  f/tr  nmointf,  in'inrifml^  (hhI  finic 
are  f/iren. 

This  is  left  as  an  exercise  for  the  student. 

SECTION    11. 

ANxrrni->  <  1.1:1  \in. 

416.  An  Annuity  Certain  is  a  sum  of  money  jxiyable  at 
the  end  of  each  year  for  a  fixed  i)eriod  of  years,  or  forever. 
In  the  one  cit.se  the  annuity  is  said  to  l>e  fprminnhh'.  in  the 
other  perpetual. 

a.  Contingent  aunuilirs  (a  inni  m  uic  iu.sui.um  r  i  an-  «»uiii>  i»a>a- 
ble  as  long  as  siH'citied  |M*rsons  remain  alive  or  for  other  uncertain 
IH^riods.  and  their  values  are  governed  by  the  laws  of  probability. 
They  cannot  therefore  bi'  taken  up  at  tliis  stage  of  the  student's 
progress. 


430  TEXT-BOOK   OF   ALGEBRA. 

The  problems  of  annuities  are  analogous  to  those  of  com- 
pound interest.  We  shall  investigate  two  cases;  viz.,  to 
find  the  amount  of  an  unpaid  annuity,  and  to  find  the  cost 
of  an  annuity. 

1.    To  find  the  amount  of  an  unpaid  annuity. 

Let  V  represent  the  amount  of  an  annuity  of  $A  which 
has  run  n  years.  Either  simple  or  compound  interest  may  be 
taken.  We  shall  use  the  latter,  and  we  will  let  R  =  1  +  r, 
where  r  is  the  rate  expressed  decimally. 

The  first  pa3anent  of  f  A  should  have  been  made  at  the 
end  of  the  first  year,  and  therefore  runs  for  n  —  1  years, 
its  amount  being  AR"~^ ;  the  next  runs  for  n  —  2  years,  its 
amount  being  AK"~"^ ;  the  third  runs  for  7i  —  3  years,  its 
amount  being  AR"""^;  and  so  on.  The  last  payment  of  $A 
being  made  at  the  end  of  the  period  bears  no  interest. 
Thus  we  have  for  the  amount  of  the  annuity, 

AR"-i  +  AR"-2  +  AR"-«  4-  .  .  .  +  A. 

But  these  amounts  form  a  geometrical  progression  whose 
ratio  is  R.  Summing  it  (406,  (3)),  we  have  for  the  value 
of  V 

^  AR-^  -  A  _  A  (R  "  -  1) 
^~     R-1      ~~~^ 

(1.    If  the  annuity  be  not  paid  until  m  years  after  the 
last  payment  is  due,  it  will  amount  to 

A  (R'*  -  1) 
r 

Example.  —  Let  it  be  required  to  find  the  amount  of  an 
annuity  of  ^10  to  run  12  years  at  the  end  of  15  years, 
interest  compounded  semi-annually  at  5  per  cent.  Substi- 
tuting in  the  formula, 

^,  _  $10  (1.025^^ -1).,^^^^ 
(1.025^)  -  1      ^  '      '' 


TOPICS   i:i:latkd  to  ki^ations.  431 

Calculating  this  by  logarithms,  starting  with  tlu*  log.  of 
l.OLC)  =  .0107.  w.*  have  log.  1.02.r*  =  0.1^508  .-.  1.025-*  = 
L.SUC). 

l(.g.  H)  =  1. 

log.  .SOG  =  i.o()(;;^ 
log.  (i.o:>r))«  =  ao(U2 

.97();"i 
h.g.  A)rAH'>  =  1^.7041' 

2.2G(>3  giving  V  =  f  184.r>2. 

(2.  Another  way  of  looking  ^t  this  problem  is  to- con- 
sider the  amount  of  the  annuity  as  an  obligation  to  be  met 
at  a  certain  date  in  the  future  and  the  payments  as  sums 
to  be  set  aside  annually  as  a  Sinh'iii'j  Fund  with  which  we 
}>ay  off  this  de])t  when  due. 
Solving  for  A.  wc  have 

A  ^''• 


R"-! 


As  an  example,  suppose  a  town  borrows  $10,(HX),  agreeing 
to  pay  it  back  at  the  end  of  15  years.  What  sum  will  have 
to  be  raised  by  taxation  every  half  year  to  meet  this  obli- 
gation,  supposing  interest  compounded  semi-annually  at  4 
per  cent  ? 

A  =  ^"<^(-"^>  =  $24r,..iO. 
1.02«>  —  1 

2.     To  find  the  rosf  of  (in  onnifif//. 

Let  I^P  be  the  cost  of  an  annuity  of  $A  to  run  n  years, 
1 


and  V 


l+r 


Then  the  present  value  of  the  first  payment  of  $A  due 
one  year  hence  is  Av ;  the  present  value  of  the  second  pay- 
ment due  two  years  hence  is  Av%  and  so  on.     The  present 


482  TEXT-BOOK  OF  ALGEBRA. 

value  of  the  last,  due  n  years  hence,  is  Ai?".     Adding,  to 
find  the  present  value  of  all,  we  have 

P  =  Ai;  +  kv"  +  Ai;3  +     .     .     .     .     -^  Ai;« 
=  A(^  +  i/^  +  t;«+ _^,,«), 

Summing  the  parenthesis  as  a  g.  p..  whose  ratio  is  v,  we 
have 

1 


1  - 


V  —  1  V  —  1  r 

(1.    If  the  annuity  \^  perpetual,  n  —  rj^ ,  and 
=  0,  (358,  2)  so  that  here  P  =  A. 


!  +  >■ 


Example.  —  Required  the  cost  of  a  perpetual  annuity  of 
$50,  supposing  the  rate  of  interest  to  be  5.  It  is  evident 
that  this  is  calculated  in  the  same  way  as  a  problem  in 
simple  interest  in  which  the  annual  interest  and  rate  are 
given  to  find  the  principal. 

(2.  To  find  the  cost  of  an  annuity  to  begin  at  the  end 
of  in  years  and  i-un  for  n  years.  Here  all  that  is  necessary 
is  to  find  the  present  worth  of  the  preceding  result  due  m 
years  hence.     This  gives 

~  r  (1  +  tY' 

{?^.  If  A  instead  of  P  is  regarded  as  unknown,  the 
al)ove  formula  gives  the  value  of  n  annual  payments  which 
will  reimburse  a  lender  for  a  loan  of  f  P. 

Example.  —  A  city  borrows  $200,000,  which  it  agrees  to 
repay  in  20  annual  installments,  interest  and  princi})al  to- 
gether. How  much  will  have  to  be  paid  each  year,  assum- 
ing a  5  per  cent  rate  for  interest  ? 


T(>ri< 


i:i:i.A  ri:i)   lo  i:(,>uati()Ns.  433 


Vr               2U()()0()  X  .05 
A  =  J  = ^£ =  {|»1()()44. 


(1  +  /•!"  1.05* 

NoTK.  —  For  more  extended  infonnalion  on  the  subject  of  inter- 
est and  annuities  certain  the  student  is  referred  to  Part  Firat  of  The 
Institute  of  Actuaries  Text-Itonk,  publislied  by  the  Laytons,  London. 


.SK(   rioN    III. 

417.  Bonds,  like  ])romissory  notes,  dniw  interest  at  stated 
intervals,  and  the  i)rinc'ii)al  itself  becomes  due  at  maturity. 

Let  p  be  the  face  of  a  bond,  n  the  number  of  years  it  has 
to  i-un,  r  the  rate  (expressed  decimally)  named  in  it,  and  / 
the  rate  the  investor  desires  to  realize  on  his  investment 
(also  regarded  as  the  current  rate  of  interest).  We  proceed 
to  investigate  the  different  forms  in  which  the  problem 
may  appear. 

1.  7V>  calculate  the  price  x  ttt  In-  imhl  for  a  bond  at  the 
time  of  its  iasue  in  order  to  realize  a  certain  per  cent  on 
the  investment. 

We  solve  this  problem  by  balancing  income  and  outgo. 
The  price  ])aid  for  the  bond  at  comj)()und  interest  at  r'  per 
cent  will  amount  at  the  time  of  maturity  to  (1  -|-  /)"x 
dollars.  The  following  amounts  come  back  to  the  investor : 
first,  the  first  interest  payment  %pr,  which  w411  therefore 
run  n  —  1  years  at  r  \^\  cent  comjKmnd  interest,  amount- 
ing  at  the  maturity  of  tlve  bond  to  %pr  (1  +  r)""^ ;  next,  the 
second  interest  j)ayment  %yr,  which  will  run  for  n  —  2 
years  amounting  to  %pr  {\  -}-  '•)"~"'^;  Ji"d  so  on  for  all  the 
payments,  the  last  being  paid  at  the  time  of  settlement. 

'  The  author  is  indebted  to  Kmory  MoClintock,  LL.l).,  F.I.A.,  actuary  of  the 
Mutual  Life  Insurance  Company  of  New  York,  for  important  practical  sug- 
gestions. 


434  TEXT-BOOK  OF  ALGEBRA. 

Then  the  face  of  the  bond  is  also  paid  at  maturity.     Hence 
we  have 

.  (1  +  >•')•'  .  X  =  pr  [(1  +  r'y-  '  +  (1  +  r'y-  ^'  +  (1  +  »■')'-" 

+    .    .    .    +l]+y^. 

Summing  the  bracket  (quantity,  which  is  a  geometrical 
progression  whose  ratio  is  (1  +  r'),  we  get, 

=  P  ('•(1  +  0"- '•  +  '■'). 

Dividing  through  by  the  coefficient  of  x. 

r\  (1  +  /)'* 

Example.  —  What  will  be  the  cost  of  a  bond  for  $1000 
payable  in  12  years  and  bearing  5  per  cent  interest,  if  the 
purchaser  desires  to  realize  4  per  cent  on  his  investment  ? 

.  =  1000  /  ^.  _  .05  -  .04  \  _  25000  ( M  -  ^^ 
.04    \  1.0412     )  \  1.04^ 

Calculating  the  value  of  -^^4^  by  logarithmic  table,  it 
equals  .00625.     Whence  x  =  f  1093.75. 

The  formula  gives  the  price  to  be  paid  for  a  bond  on  the 
assumption  that  all  interest  payments  are  made  annually. 
But  they  are  usually  paid  oftener  than  once  a  year. 

2.  If  the  ifiterest  payments  are'7nade  q  times  instead  of 
once  a  year. 

The  period  rate  will  now  be  -,  and  the  terms  in  the 
amount  series  will  be 

r  ,x"-i    pr  ,.„_!    pr  ,.«_•!  r 


TOPICS   RELATED  TO   EQUATIONS.  435 

Hciv  the  ratio  is  (1  -h  r)i .     Hence  we  have      (406,  167), 

L  ,^  (1  +  /yl  _  y  J 

...  a-  =  t  \''  ^^  -^'''y^  +  y  (^  -t-  ''')\  -  w  +  ^-^l 

This  and  the  preceding  result  were  calculated  on  a  basis 
of  annual  compound  interest,  l^ut  regular  investors  expect 
to  compound  their  interest  semi-annually.  Tables  of  bond 
values  are  usually  constructed  in  accordance  with  this 
practice. 

3.  If  the  hiteresf  payments  are  made  q  times  a  t/ear  ami 
the  interest  is  compounded  semi-annually. 

In  order  to  compound  the  interest  semi-annually,  instead 
ol  (1  H-  /•),  we  substitute  (1  -|-  ^Y^  1  -f-  r,.  This  latter  (/•,), 
is  often  called  the  effective  annual  rate  corresponding  to  the 
nominal  rate  /. 

^  _  /T  r  (1  4-  r,y  +  y  (1  +  r,)T  ^  (y  +  r)! 

Example.  —  Required  the  present  value  of  a  mnnici})al 
bond  for  $2CK)f)  to  run  for  20  years  having  six  per  cent 
coupons,  payable  (juarterly,  to  net  purchaser  4^^  per  cent. 

Here  /  =  .045 ;  whence  r,  =  (1.0225)'^  -  1  =  .0455  -f  ; 
log.  1.0455  =  .0193;  .0193  X  20  =  ..3800,  giving  (1  -f  /•,)- 

-  2.4.322 ;  \  of  .0193  =  .0048,  giving  (1  +  r,)i  .  .  .  =  1  (H 12. 

X  =  ?!?^  r.06  X  2.4322  +  4  X  1.0112  -  4.0()1 
4     [_  2.4322  X. 0112  J 

500  X  .1307 
"  2.4322  X  .0112 


436                           TEXT-BOOK  OF    ALGEBRA. 

log.     500  =  2.6990  log.  2.4322  =  0.3860 

log.  .1309  =  1.1163  log.    .0112  =  2.0492 

1.8153  2.4352 

2.4352 


log.         X  =  3.3801 
.-.  x  =  2400. 

4.  If  interest  pay  merits  have  been  made  on  a  bond  like  a 
promissory  note,  it  virtually  becomes  a  neiv  bond,  to  run  for 
the  remainiiig  time,  and  a  jyrospective  purchaser  may  calcu- 
late from  this  standpoint.  If  bought  at  some  time  bettveen 
interest  payments,  the  discount  is  from  the  time  of  sale  to 
maturity.  Thus,  instead  of  n  as  the  exponent  of  (1  -|-  /), 
or  (1  +  ^f)?  ^'^  the  denominator  of  the  value  of  x,  the  exact 
discount  interval  must  be  substituted. 

Example.  —  Let  it  be  required  to  find  the  present  value 
of  a  bond  for  $1000,  drawing  7  per  cent  in|:erest,  payable 
semi-annually,  dated  Jan.  1st,  1895,  and  due  Jan.  1st,  1910, 
on  June  18th,  1898,  supposing  the  purchaser  is  willing  to 
take  5  per  cent  for  his  money  (to  be  compounded  semi- 
annually). 

Here  we  may  calculate  the  bond  as  if  it  had  been  issued 
on  Jan.  1st,  1898.  It  will  therefore  have  but  12  years  to 
run.  Also  the  exponent  of  (1  +  r^  in  the  denominator  will 
be  11^8^.     The  effective  rate,  r,  =  .050625. 

Log.  1.0506  =  .0214;  .0214  X  12  =  .2568 ;  (1  +  r^^ 
=  1.8062 ;  .0214  X  11-1%  =  0.2466 ;  \  of  .0214 
=    .0107;  (1  +  r,)7  =  1.025. 
_  1000  (.07  X  1.8062  -f  2  X  1.025  -  2.07)  _ 
^~  2  X  1.0506"A  X  .025  ~  *^^  ^^• 

5.  To  calculate  the  rate  of  interest  tuhich  can  be  realized 
when  the  cost  of  a  bond  is  given. 

In  order  to  make  this  problem  soluble  by  a  formula,  we 
distinguish  between  the  current  rate  of   interest  and  the 


Tones  i;i;i.A  iKD  so  equations.  487 

rate  to  be  realized,  iiy  tlie  current  rate  of  interest  we  mean 
the  rate  at  which  the  interest  payments  can  be  promptly 
re-invested.  Let  us  denote  this  rate  by  r^.  Then  the  for- 
mula in  2,  above,  becomes 


P  \r(l  +  r,)  -  4-  g  (1  +  Te)^  -  (y  4-  r)\ 


Solving  for  /,  which  is  now  regarded  as  unknown, 

Example. — A  $500  bond  bears  8^  interest  payable 
semi-annually,  and  is  due  18  years  hence.  If  the  current 
rate  of  interest  is  3J,  what  will  a  purchaser  realize  who 
Ijays  .f  800  for  it  ? 

Here  r,  =  .0353. 
Log.  1.0353  =  .0150;  .0150  X  18  =  .1>700;  (1  +  r,)-  =  1.862; 
\  <,f  .{)\m  =  .(K)75;  (1  +  r,)l  =  1.0175 

Remark  ox  the  Last  Case.  —  Properly  speaking  only  two  rates 
of  interest  are  recognized  in  bond  calculations ;  that  named  in  the  bond 
itself,  and  the  rate  to  be  realized  by  the  purchaser.  However,  we 
may  substitute  a  probable  value  of  the  latter  on  the  right  side  of  the 
al>ove  equation,  and  thus  find  an  approximate  value  of  the  rate  to 
lie  realized.  This  answer  could  be  used  in  a  second  similar  calcula- 
tion to  find  a  second  approximation. 

However,  a  simpler  and  preferable  method  of  solution  is  to 
assume  two  rates  for  r«  (one  too  large  and  the  other  too  small)  in 
the  formula  in  3,  and  calculate  the  corresponding  values  of  x.  Then 
the  desired  value  of  r,  may  be  found  from  the  given  value  of  x  l>y 
interpolation. 

418.   Exercise  in  Interest,  Annuities,  and  Bond  Calculations. 
1.   What  will  $1  amount  to  in  20  years  at  5  per  cent, 
interest  beinj^  compounded  semi-annually  ? 


438  TEXT-BOOK    OF   ALGEBRA. 

2.  What  sum  of  money  will  amount  to  ^1000  in  8  years 
and  4  months  at  4  per  cent  compound  interest,  interest 
being  compounded  quarterly  ? 

3.  At  what  rate  per  cent  per  annum  will  $500  amount  to 
^760.17  in  8  years  and  6  months,  the  interest  being  com- 
pounded semi-annually  ? 

4.  How  many  years  will  it  take  a  note  for  $100,  bearing 
4  per  cent  interest  payable  quarterly,  to  amount  to  $150  ? 

5.  In  how  man}^  years  will  a  note,  whose  face  is  $300, 
amount  to  $400,  allowing  8  per  cent,  payable  quarterly  ? 

6.  In  how  many  years  will  $967.80  amount  to  $1269.00 
at  5  per  cent  compound  interest,  the  interest  being  com- 
pounded semi-annually  ? 

7.  The  population  of  the  United  States  in  1880  was,  in 
round  numbers,  50,000,000;  in  1890,  62,500,000.  What 
was  the  annual  gain  per  cent  in  this  decade,  supposing  it  to 
have  been  the  same  throughout  ? 

8.  At  the  same  rate  of  growth  how  many  years  would  it 
require  for  the  population  to  reach  100,000,000  ? 

9.  Allowing  6  per  cent  interest,  what  will  be  the  cost  of 
a  perpetual  annuity  of  $500  ? 

10.  Calculate  the  annual  interest  and  amount  of  a  note 
for  $500  which  had  remained  unpaid  5  years  and  3  months. 
Rate  named  6  per  cent. 

11.  A  gentleman  wishing  to  establish  a  free  scholarship 
of  $300  to  be  in  force  for  fifty  years  desires  to  know  how 
much  it  will  cost  at  a  5  per  cent  rate  of  interest.  What 
will  he  have  to  pay  ? 

12.  An  annuity  of  $20  ran  for  thirty  years.  How  much 
would  it  have  amounted  to  in  all  had  the  payments  been 
withheld  until  the  last  became  due '/  Assume  the  rate  to 
be  6  per  cent. 


TOPICS    KKLATKl)    TO    IX^rATlONS.  4od 

13.  What  will  an  annuity  of  $100  cost  to  begin  in  10 
years  and  run  for  25  ?     Rate  4  per  cent. 

Note.  —  If  any  annuity  begin,  say  Jan.  Ist,  11K)1,  tlie  lirst  pay- 
ment on  it  will  not  be  made  till  Jan.  1st,  1902. 

14.  A  town  borrowed  $18,000,  agreeing  to  repay  it  in  25 
annual  installments.  What  sum  had  to  be  raised  annually, 
if  5  per  cent  was  the  rate  named  in  the  contract  ? 

15.  An  annuity  of  $50  was  to  be  paid  semi-annually,  $25 
every  six  months,  and  to  run  for  eight  years.  Not  being 
paid  until  eight  years  and  six  months  had  elapsed,  and  the 
current  rate  of  interest  being  G  per  cent,  what  was  the 
amount  to  l)e  paid  ? 

16.  A  gentleman  wishes  to  purchase  a  $1000  bond  bear- 
ing 7  per  cent  interest,  payable  quarterly,  due  in  ten  years, 
so  as  to  realize  5  per  cent  on  his  investment.  What  can  he 
afford  to  pay  for  the  bond  if  he  compound  his  interest  semi- 
annually ? 

17.  A  $1,000  bond  bears  6  per  cent  semi-annual  interest 
and  is  to  run  12  years.  It  is  offered  at  $1025.  What  an- 
nual rate  will  be  realized  from  it.  by  a  purchaser,  allowing 
s  'nii-annual  interest? 

SuooEHTiox.  —  Assume  .05 J  as  the  value  of  r«. 

18.  On  a  bond  for  $1(MM)  payable  by  6  per  cent  semi- 
annual coupons,  and  which  was  to  run  24  years,  the  sixth 
payment  of  interest  has  just  been  made.  What  sum  can  be 
paid  for  it  to  net  the  purchaser  4^  per  cent,  computing  all 
the  interest  semi-annually  ? 

19.  A  town  issued  a  series  of  12  bonds  of  $500  each  dated 
May  1st,  1802,  and  bearing  6  per  cent  interest  payable  semi- 
annually. The  first  was  to  become  due  May  1st,  1900,  the 
second  May  1st,  1901,  and  so  on,  the  last  to  become  due  May 
1st,  1911.     What  can  a  purchaser  afford  to  pay  for,  say,  the 


440  TEXT-BOOK   OF   AI.GEBKA. 

first  and  last  of  the  series  on  Sept.  15,  1892  desiring  to 
realize  o  ^,  interest  compounded  annually  ? 

20.  A  bond  for  f  2000  bears  5  per  cent  semi-annual  in- 
terest and  is  to  run  30  years.  What  rate  (allowing  semi- 
annual compound  interest)  will  be  realized  if  it  be  purchased 
3  months  after  its  issue  for  f  1911  ? 

SuGGESTiox,  —  Assume  .05|^  as  the  vahie  of  Vg. 


ANSWERS. 


ART. 

18.    2.  +;");  3.  "7;  4.  -'JO;  6.   Mr>;  6.  "l.'J:  7.  +4:  8.  "  :i : 
9.  +1';   10.  -r>;   11.  "41  ;   12.  -4(;. 

28.    1.  (.'}.  -f  11<S;   (4.   -  ;U1:  2.  {:;.  -  40;   O").  -  7. 

30.  1.  :V27',  2.  -1)5;  3.  (VJ-A:  4.  -  1*>1^7:  5.  -  :\()\;i; 
6.  r>i).228  ;  7.  +  34  ets. 

31.  9.-7;   10.  'J'A:   11.  -  ."iC* :   12.  —  r>S'J :   13.  —  1^001  ; 
14.  —  IT);""). 

33.    1.  -f  221^2;  2.  -1-19:   3.  —'.Ml;  4.  —  .VUf; ;  5.  -{-  VM\ 
6.  +  17°. 

37.    1.  -  KJ:  2.  -h  LMC;  3.  -  r.7  :  4.  —  42;  5.  -f-  I'SS  : 
6.    -  S«. 

41.  1.  -  \i\r>n  :  2.  —  .SS:  3.  r)7<i(> ;  4.  -  .".',:  5.  .(MrjKi; 
6.  -  r.-.. 

42.  2.  (1.    2r)li;   (L\  jr>{\',   {^:\.  Aim>:   (4.  -  lai; 
(5.  -  2744  ;  ((>.  12.S.LM  ;  (7.  —  LMl)7. 

44.    1.  _  r,;  2.  -  T):  3.  —  10:  4.  —  1  :  5.  —  .77."); 
6.  -.ll.';;  7.  .().S2. 

46.  1.  i  (),  i  13,  i  7,  i  i:>,  J-  1 1 :  2.  2.  -  :;.  -  4.  r,, 

-7;  3.  ^2,^3,^4;  4.  i  2,  i  :;. 

80.    1.  '/  -h  ^'  +  '• :  2.  2  A ;  3.  </  -f-  -  :  4.  ^^  —  r» ; 

r 

r>.  .,»-h.S/'^-f- W:  6.  ^'^    ,    4A_^/ 
A   ^  ;5  ..        24 
82.    1.  1  2  :  2.  1  1  1  :  3.  -  178 :  4.  4  ;  6.^  12 ;  6.  i  22. 

or  ^tz  1<>.  <>r  ik  <*»•  <•»*  0. 

441 


442  TFXT-BOOK   OF   ALGEBRA. 

83.    1.  144 ;  2.  840 ;  3.  198 ;  4.  200. 

85.    2.  a;  =  7;  3.  12;  4.  6;  5.  10 ;  6.  5;  7.  4 ;  8.  5 ; 

12.  24,  16  ft.;  13.  $7,  70,  28;  14.  50;  15.  9,  14,  27;   16.  45 
17.  4,  6,  9  cts. 

89.    1.  (1.  9a;  (2.  7am;  (3.  0;  (4.  -7a^;  (5.  -2m^n; 
(6.  2  axt/ ;  2.  20  «i^  —  2  a^^,  ^  3.  a-  -  (w/  +  3  //  -  2 ; 

4.  5a-\-Ab  —  Sc  —  e;  5.  6 ;  6.  0 ; 

7.  6  ab  -\-  7  ax^  —  4  a^x  -\-  ax^  —  5  a%', 

8.  5  cc^  +  4  ic^  +  3  o;-^  —  4  a;  —  9. 

93.    1.  (1.  2a;  (2.  5b',  (3.  -  17c;  (4.   -x'-,  (5.  31a%^: 
(6.  a^b  —  ab^  ;   (7.  —  15  ad^  ;   (8.  24  a^b^  ;   (9.    6  m  ; 

2.  a?^  _  8  ^2^2  +  12  ;  3.  2  ax  -  7  ^*?/  +  8  c^ ; 

4.19ab  —  2c  —  7d  —  2x^-\-3x'^  —  x  —  l',   5.  2  ax  -{- 2  x'^ : 
6.  a-3  —  2  .t2^v  —  2  a-V'  +  ?/' ;  7.  2  .ry  —  (;  rz  +  6  w  : 
8.  2  a^  -  2  a'c  +  Sac'^. 

96.  2.  6  (a-\-b)  ;  3.  -  2  (;r  +  z)  ;  4.  16  (a  +  by  ■ 

5.  5(x-\-  y)  -\-  (x  —  //)  ;  6.  -  9  {a  +  2  bY  -  8  (a  -  m). 

97.  2.  (a  +  3)  ic?/ ; 

3.  («  +  ^  +  ^  +  //)  ^  +  (^^  +  d  -h/4-  A)  2/; 

4.  (a  +  2  c  +  4  cZ)  £r. 

98.  2.  (a  _  Z,)  (.r  +  7/  +  ;.)  ; 

3.  {a  +  m)  (.X  +  2/)  +  (^  -  n)  (ir  -  y). 

101.  1.  3a;2-2y  +  2a;- 1;  2.  2x^'dy;  Z.4.xy; 
4.2b  -  Ac;  5.  0 ;  6.  3  m. 

102.  2.  3a  +  3c;3.  5a  —  6a;;  4.  a-^  —  »•;  b.  2a  —  Ax; 

6.  _  rt  +  8  ^^  +  7  a-  —  6  ?/  -  2 ;  7.  12  a;  —  8  y. 

103.  1.  (a  +  Z,)  +  (,  +  ^)  ; 

2.  (a^  +  2  aZ*  +  Z/'^)  -  (c^  -f-  2  c^  +  d^)  ;  3.  (a  +  Z<)  +  c. 
108.    1.  —16;  2.  -12a2;  3.    45  ^/i^^^.  4   _24a^»c; 

5.  6  a^^/^a:^  ;  6.    —  2  aa;^?/  ;  7.  14  a%c  ;  8.  6  a^^  ;  9.  a%'^  ; 
10.  4  a;^?/»  ;  11.  —  20  a%-^x^y  ;  12.  a;^^^,-^ ;  13.  2  a;"*  +  ^ ; 
14.  —?/«  +  *. 


ANSWERS.  443 

112.    1.  —  288  rtV.» ;  2.  8  ac  -f  \2  be ; 

3.  _  ^  ahxy  -f  ^  acx'^  —  21  nx  ; 

4.  78  a^mVy  —  60  a*//*'///V/  +  •'<>  "  '//'*^y ; 

6.  m*  -|-  m»  +  m«  -|-  /wc  ;  6.  —  14  ar  -f  28  />6"  -|-  42  c^ ; 

7.  2  «2^,c  —  2  a^»-^c  —  2  a^.c'^ ; 

8.  —^(i*b^-\-2\a^b*-\-A0a%^\  9.  o-'^  +  5ar  +  6; 

10.  x^-\-Qx  -  10 ;  11.  />^2  _j_  ^23.2  _  ^,2^2  _  ,^2^2 . 

12.  .!•<  —  y* ;  13.  ar»  +  ar-  -  4  a-  +  80;  14.  a-^  —  14  ar  +  45 ; 

15.  ./-  4-  2  ^/.r  -f-  a"-^ ;   16.    m'^  +  («  +  b)  m  +  <//; ; 
19.  .S  ./•-  -(-  4  J-//  -f-  //2  —  2  y^;  —  3  5;^ 

115.  3.  49  +  70  4-  25  =  144  =  12=^ ;  7.  9  «^  +  12  nb  +  4  b' ; 

8.  25  i'W'  -  40  tW  4-  16  c'-^rf* ;  9.  —  +  — '  +  — ; 

4    ^     3     ^  9 

16.  </-^  —  2a^b»  -\-b^y. 

116.  7.  (1.  x^  +  7  a-  -f  12  ;   (2.  .T'^  +  15  X  +  54  ; 

(4.  y^  —  Ax  —  O]  (10.  ar=^  -h  X  —  240;  9.  (1.  x'  +  24  x  4-  14.S ; 

(5.  a'  4-  ^-^  4-  1  4-  ^  rti  4-  2  rt  4-  2  ^  ; 

(0.  a^4-^'-'4-9-2«^4-0a  — 6^*; 

(9.  9  m'^  -{-  n^-^  4p^  -\-l-\-Q  mn  4-  12  mjo  4-  (>  m  4-  4  np 

4-  2  «  +  4/> ;  (m  a!»  4-  6  xhj  4.  12  a*//*  4-  8  //». 

117.  1.  (3.  (2).  9  a-y,  Z»-^S  9  c^x\  4  «'^//V,  K;  r/«^;»V^  «'^"', 
a-^'';  (3).  8«W4»,  -'27«V>»,  —  ^a%hnhi\  34.S;A/V.  ff""^*"' 
(4).  768  x'\  -  32  x"//*,  a^^x'K  pY'.  -  2  A',  o^\ 

121.    1.  27;  2.  —4;  3.  24^/-/.:  4.  ./-./•-:  5.  -4oVr:  6.  5 .v V : 

7.   _  1  ,.y»r-;  8.  -  ^^  ;   10.  '/""V; :   12.  ;;  (^^  -  />)•'. 
2  .r// 

127.    2.  3  m^  (a  4-  c)-*;  3.  3  a-^" ;  4.  ff'"'; 

6.  —  5  /w/'  —  4  /« //  4-  2  // ;  6.  3  //»  —  4  «//•' ; 

7.  —  3  y>r.s»  4- 6  jor<2  4-7  ra--';  8.  2a4-4c4-6i; 

9.  3  (a  -  ^)<  —  2  (a  —  i)='  4-  (a  —  b);  10.  2ar  4-  3//; 

11.  c  —  d;  12.  3  —  a;  13.  2  a^  -{- 3  ab  -{- b^;  14.  a:  4- // ; 

16.  a;*— .^  4-  ^;  16.  3a:«  — x  4-  2;  17.  a*  -\-  a^x  4-  wa-*  4-  a-'. 


444  TEXT-BOOK   OF  ALGEBRA. 

128.  3.  (1.  1 ;  (2.  1 ;  (4.  81,  27,  9,  3,  1,  ^,  i,  ^^,  ^  ; 

129.  1.  (3.  (1)  i4«&2;  (2)  i3«^.^;   (3)  2^/;   (4)  ^4; 
(5)  -  3  7^i ;  (6)  -  2  ac' ;  (7)  ^  25^/^'  5  (^)  i  3  ic^ 

131.    1.  a  +  ^^ ;  2.  (wi,-^  +  wi^i  +  71^)  ; 
3.  52  +  5  X  2  +  2^  =  39;  4.  1  +  y  +  t/^; 
5.  (2  a  +  3  b)  (4  a^  _  0  ^^,  _|_  9  ^,2^) .  q  ^12  _  ^6^12  _^  ^24. 
7.  (J^^  -f  ?/;s^)  (x'  -  yz^)  ;  8.  (a*  -  /y^)  («,8  _|_  ^4^3  ^  ^6^  . 

11.  (4^4-10  h)  (16  ^;^  -  40  aZ»  +  100  li")  ; 

12.  (2  n  -  h)  (32  rA^  +  .  .  .  4-  />5),  (2  ^t  4-  ^.) 

(32  ^/^  —   .   .   .   —  h^)  ;  14.  (m2  +  n^)  (m^  —  m^n^  -\-  n'^)  ; 
17  —  20.     Not  divisible. 

133.  1.  (2.  2  X  7m  -  m-  m-  71-, 

3.  7  X  13  ((  ■  a  •  a  •  a  '  b  •  c  •  c  •  c ; 

4.  33  (^6  4-  b)  (a  -h  /v)  (^r  -\- b) -,  2.  (2.  5  a;  (4  ic^  _  9  ,^2) . 
3.  ./'-^  (12  r^  —  3  b  -\-  1) ;  (4.  xY  (^'^  —  a-^  4-  1)  ; 

r>.  7  br'x  (2  6'2  -  3  bf  +  //^) ;   (k  3  //f2  (2  x  -  T)  c  —  b). 

134.  1.  (2.  (^/.-^  4-  f''')  ("  4-  ^>:)  {^(  -  b)  ; 

(3.  {nb  +  c^)  (r///  -  cd)  ;  2.  (2.  (./;  -  v/)  (.r^  4"  •''//  4"  //')  ; 
(3.  (.r-2)(a;-^4-2.r  +  4); 

3.  (2.  (x  4-  ;//)  (./-^  —  r^//  4-  ..•>//'  —  .''-//•■'  4-  •'■.'/'  -  r)^  (.''•  —  !/) 

(x''  4-  xhj  4-  xhf  4-  J-!/'  4-  .z-,//^  4-  //'v ; 

(3.  (a  4-  2)  {a^  —  2  r/-^  4-  4  a'  -  8  ^//^  _|_  k;  ^,  _  ;>2).  ^,,  _  i>j 
(«5  4-  2  a^  4-  4  (i^  4-  8  <r-^  +  If)  a  4-  32)  ; 

4.  (2.  (.r  4-  3  //)  (./-  -  3  .r//  +  9  y'). 

135.  1.  (2.  (,-  4-  5;  (..  4-  ^)  =  {X  +  ^)^  (3.  (.>•  4-  4)^ 
(4.  (..  4-  6  ^)^  2.  (2.  (^  4-  3)  0.  4-  2) ;  (3.  (^  4-  ^)  (^  +  <^)  ^ 
(9.  {rs  -f  5  .^)  (I's  4-  18  .^)  ;   (12.  (./•  -  5)  (.r  -  4) ; 

( 19.  {n  4-  5)  {a  -  3)  ;   (20.  (x  -  (>)  (x  +  5)  ; 
3.  (2.  C^x  4-  2)  (3a^  4-  1)  5  (3.  (3 a^  4-  7)  (X  4-  2) ; 
(4.  (4.r-l)  (.r4-3);  (5.  (9.7-4-1)  (.r  4- 7) ; 
(0.  (3  x-2  y)  (x+Aij)',   (7.  (2  .r  4-  1 )  (.r  -  1 ) ; 
(8.  (3  ;/•  4-  2)  (./'  -  7)  ;  (9.  (2  r  -  ./)  (r  -  (J ./)  ; 


ANSWERS.  44.") 

(10.  (2  m  +  ij)  (//i  -  2  y) ;  (11.  (12  x  +  5)  (j-  -  3)  ; 
(12.  (15  «  +  1)  («  -  15) ;  ( l:5.  (.S  a-  -  4  y)  (8  .r  +  y)  ; 

5.  (2.  (5  a-2  -  .r  -  12)  (r,  ./•  +  1)  ;  (3.  (4  a-'-^  -  2  ^-  -  5) 
(2x-5). 

136.  1.  (a  (2)  (a  +  1)»;  (3).(4^»  +  1/;  (4)  (2;r  -  5y)»; 
(^.■-)  (<;  X-  -  y)»;  2.  (2.  (2  a  +  :U  -  c)  (2  tt  -  3  ^  4-  r)  ; 

(3.  (2  a  +  3  ^»  -  2c)  (2  «  -  3  ^»  +  2  c)  ; 

(4.  (2a  +  c  +  3/0(2a  +  r-3^); 

(5.  (/  +  m  —  7i)  (/  -  m  -f  w)  ;  (6.  («  -}-  ^  +  ^0  («  -\- 1>  -  •') ; 

(7.  (a  -  i  +  1)  (a  -  ft  -  1)  ;  (8.  (4  a'  -  4  ft^)  (2  a'  +  2  ft^ ; 

(19.  (4  a-2  +  3  .r//  -  y^)  (4  x^  -?ixy  -  f), 

or  (Ax'  +  5a-y  +  y^)  (4x^  -  5«//  +  y'^)  ; 

(20.  (3  x«  +  2  xy  +  7  y«)  (3  x«  -  2  .ry  +  7  //-) ; 

(24.  {2x^-{-(Sxz-\-^z'^  (2ir*-6a-«  +  95;2); 

3.  (3.  (1)  {a  J^h){e-d)',  (2)  {a'  +  1)  («  +  1); 

(3)  (ar  -3)  (y  +  2);  (4)  (3a; -2y)  (2a -7y); 

(5)  {m  _  3  w)  (9  a  —  4  ft)  ;  (6)  (m  +  »«)  („<2  _  w)  ; 

(7)  (x^-,/)  (x-y);  4.  (3.  (m^  +  4  7/^  +  1)  (w+  1); 

(6.  (a-2  —  4  a:  +  (>)  (4  a-  -  5). 

137.  2.  (2.  («-|-^,-|-c  + J)  (aJ^b-\-c  -d)', 
(3.  (2  7/^  -  3  7*  +/>  -  2  *7)  (2  w  —  3  w  -  ;>  +  2  y)  ; 
3.  (2.  («  -  3  ft  +  5  r)2. 

142.    1.  3  a- ;  2.  2  «  ;  3.  b'c ;  4.  4  x^y'u^  ;  5.  2  («  +  ft)  ; 

6.  7  xy ;  7.  2  a'-^afy  (3  x  —  y)\  8.  a-  —  y ;  9.  Triine  to  eacli 
other;  10.3a(a;  +  4);  11.3(a;-l);  12.  ./'3_r>;  13.  ./"" -f-fl; 
14.  r  (a  —  ft): 

148.    1.  31;  2.  12G;  3.  2;  4.  a;  +  7 ;  5.  x  -  2y; 
6.  2  a  -I-  3  a; ;  7.  c  (a^  —  ?»)  ;  8.  3  a*  +  4  ^/ ;  9.  a-  (ar  —  1)  ; 
10.  a-  (a;  —  2) ;  11.  3  y  -  7 ;  12.  ax  -  fty ;  13.  2  a:  -|-  1 ; 
14.  x'  (3  a-  +  2)  ;  15.  x^  -  5  a:  -f  ^>)  ;  16.  («  +  ^f- 

153.    1.  (>48 ;  2.  720 ;  3.  21(>0 ;  4.  252  axh/\  5.  x^  (a  +  a  ) ; 
6.  X  (x^  -  1)  ;  7.  3(>  aVA'^d' ;  8.  a-*  -  y^ ;  9.  «*ftVa-* ; 
10.  SOx'Yz';  11.  54a»ftV;  12.  72  aW;  13.  210^2/,^; 
14.  12  xy^  (x^  —  y«)  ;  15.  8  (1  -  x^)  ;  16.  ab  (a  +  ft)  ; 


446 


TEXT-]JOOK    OF    ALGEJiltA. 


17.  12  x^  (x^  +  2)  ;   18.  xi/  (4  x^  —  1) ;   19.  xhf  {x^  —  /)-^; 
20.  24  ah  {ii^  —  l^)  I  21.  abc  (x  —  ft)  {x  —  h)  (x  -  c). 

160.    2.  i,  ^L,  .  ;   3.  A,  ,^\,  ;  4.  :^ ;   5.  7  ^// ; 

12.  t.   13.        ^'      . 

c-  '  «'^  +  Z*-^ 

162.    2.  31,  71^,  5H,  12ftV;  3.  Z>  +  ^;  4.  x  +  y;  5.  x^A 


6.  2  — 


2x^ 


9.  2  </  -I 


'Z  x'  —  X 


;  7.  a'x  +  x' f^-^  ;  8.  x  +  — ^^ 

+   1  6t^  —  X^  X  -\-   ^6 

Ix 


,  ;  10.  2  a^'-^  -  '6  a'b^  + 
«  —  t*  11  a^ 


164.   2.  ^--A-  +  -iL;  3.1  +  1  +  1  +  1. 
o  Or      10  ^/r       'Sab         d       a       b       c 

166.    2.  5«^-^e-2;  3.  ab-h-hf-  4.  ^a-^ccZ-S; 

168.  3.  w,  ¥,  y ;  4.  ^^-^^-^.  5^^  +  ^^ 


b. ,  7.  

6x 


a  11  0:;''^  +  5 

a  -\-  b^  5x 


ab^d 


a^  -  ^-^ 


9  ac     5  ^'•■^ 


_   adf    bcf    bde  ^  ^     6  a'        3  ^»'^       10^  . 


^>r//-'  bdf  bdf  '     *  30  abc   30  «^»c'  30  a^c  ' 


7. 


15  (fiTia?     4  nx     10  a^ 


lOa'^/i'  10  aV   10  ahi  ' 


g  567  a     98  Z>     198  (/a;  882  {a  +  &) 
■  504  X   504  ic'    504  x  '       504  ir       ' 

175.     2.  2A  ;  3.  t'^^  -1  ;   4.  |o  .^  ;    5.  ^^- +  ^^+  ^^^  ; 
10  /></// 


ANSWERS.  447 


11.  S  ./•  _   .;•    a  ;    12.  1)  ^;-  +  "'"•'""". 

177    1   .0.,  .  0  i^iLnil.  3  JL^.  4      ^(^^'  +  ^^^0     . 

6.  -1^  ;  6.  y-^:  7.  ^"^>"+^   8.  .+l>.  +  ^!-^^ 


•   10.  ■' ;   11.    -= ;   12. 


17.  tF^^;  21.0. 
ft^  —  J"* 

179.    2.  ./• :  3.  ;  4.  —  .)  oh  ;   o.    — '-  ;  6. : 

nl  \br  1  —  x 

7.  — -^  ;  8.  ///-  -  /r ;  9. — —  ;  10.  -^— ^  ; 

«    '        3  (a;  -  y)  '         X  ^    '         10  a  -  5  «■' 

181.    2.  -  2^^  3.  -  J!il=  4.  »*;  5.  i^N  6.  ^V^  : 

;!r/  4.t*  «  4'/  1(1. r-' 

7.i^';8.  -L-;9.fL±*;  10.       ^ 


7>  (r  —  1         X  -{-  //  (f  -|-  b 


183.  2.  ^_(^^-±i);  3.  H<^^  +  f^),  4.  ?li:^^  5.' 


rtft*-' 


^  (y  —  ^')         «  (^^-^P  —  «)  1<>  ^'^''  4 

7. 


*^    ;8. -^V;9-^^^%^;  io.-L_; 


a  -f-  bi/         m-\-bn         a  (m^  -j-  1)  ///  -f-  1 

•l0(3x-l)* 

209.    2.   I  :  3.  1();  4.  2,7^  ;  5.  — ;  6.  —  :  7.  7  :  8.  — 

a  .H  (I  n 

9.  2H  ;    10.  ,';  ;  17.  ^?-±4±/;  19.  a  -  /. :  21.  4. 

"  -\-b-\-  c 

211.    3.  1  :  4.  ;ij;  5.  -^^ ;  6.  2;  7. 


m  — a  a  +  6  +  c 


44S  TEXT-BOOK    OF   ALGEBRA. 

8.  ^  ""  "       ;   10.  1 ;   11.  1 ;   12.  4;   13.  1. 

m  -\-  0  ~{-p 

213.  2.  11^;  3.  7;  4.  7;  5.  275§§  ;  6.  420;  7.  7 ;  8.  ^ ; 
9.1;  10..;   12.^. 

214.  2.  6 ;  3.  25  ;  4.  6. 

216.    1.4^;  2.  4;  3.45;  4.  ^p^ ;  5.4;  6.  7;  7.  12;  8.  7; 

9.  i  ;   10.  3  ;   12.        '''''  ~  ''^'      - ;   17.  ,\  ;  28.  4 ;  33.  -  9; 

a-\-h  —  111  —  n 

49.  a  =  (iLZlM,  h  =  d{a-\-c)^ 
d  —  h  a-\-d 

224.    4.  7  ;  5.  1()(>,  75 ;  6.  60  ;  7.  58^2^  ;  8.  6 ;  9.  10  ; 
11.  25 ;  12.  22,  10  ;  13.  6,  8 ;  14.  14,  12 ;  15.  15,  5 ; 
16.  3,  9,  15,  21  yrs. ;  17.  21,  7,  14  cts. ;  18.  25,  75 ; 
19.  20  of  $100  .  .  .  1280  of  $1 ;  20.  34, 17  gals. ;  21.  35,  90 ; 
22.  $42,  28,  and  18  ;  23.  175  mi. ;  24.  30  ft. ;  25.  ^  oz.  ; 
26.  5;  27.  54;  29.  54;  30.  143^1  mi.;  32.  $12000; 
33.  $25200,  $3000;  34.  lOfc  ;  35.  30;  36.  $45;  37.  3; 
38.  37^  ft. ;  40.  10  da.  ;   41.  4  min.  ;   42.  40  miu. ;  43.  21 ; 
44.  12i§  min.  ;  45.  60  ;  46.  120. 

230.  3.  X  =  2,  ?/  =  1,  or  2,  1 ;  4.  3,  5 ;  5.  8,  2 ;  6.  25,  15 ; 
7.  _  1 0,  —  60  ;  8.  1,  3 ;  9.  8,  6 ;  10.  12,  8  ;  11.  6,  15  ; 

232.    2.  8,  7;  3.  8.  1 ;  4.  6,  3;  5.  21,  35;  6.  5,  5:  7.  3,  5. 

234.    4.  5,  2  ;  5.  20, 15 ;  6.  4,  6  ;  7.  —  6,  80  ;  8.  20,  60 ; 

9.  3,  5 ;  10.  7\,  5 ;  11.  3,  4 ;  12.  9.\,  7  ;  13.  .4,  .1 ; 

14.  ^Jl^zJ^\  P  —  y ;  15.  —       ^"^         _!!!!!_, 
a  (h  —  f/)'  b  —  d^  (m  -f-  an)^  w  -\-  an 

236.    2.  30,  9  ;  3.  33i^,  21]  ;  4.  6.3.  39.2  ; 

a  -  li'  ah  —  1 


5. 


a^  —  i;  a^  —  h 

237.    2.  15.  3  ;  3.  11,  -  1  ;  4.  2,  —  1 ;  5.  7,  11. 


A.NSWKijs.  449 

239.  1.  (>,  3;  2.  10,  7;  3.  .4.  .1  ;  4.  20,  4;  5.  57.  \n:\: 
6.  (5,  10;  7.  12,  30  ;  8.  2,  (^  ;  9.  1,4;   10.  19,  3  ; 

^^   ;<  (r  -  d)   m{c-d),  j2.  r>.  4  ;  13.  -^'^  +  ^\  '^^JJ^^-Z^, 

mi   -f-  hnt     on  -f-  />///  7  —  2  b         7  —  2  A 

240.  2.  90,  (JO  fts. ;  3.  7142Sr),  142Sr>7;  4.    .1i;420,  $200; 
6.  180,  145;   6.  92800000,  07100000  lui.  ;  7.  $12,  10; 

8.  $2r>0,  320;  9.  2.322,  11.03. 

244.  1.  1,  2,  3;  2.  7,  8,  9;  3.  VKS.7,  05.4,  32.1 ;  4.  24,  9,  5; 

6.  (•),  2,  1  :  6.  3,  9,  15;  7.  4.5,  10.8,  11.7;  8.  7,  4,  3;  9.  /,  a, 

'"'  "'  *''  "  '       ■       (a'  4-  A'*)  r      '       (a^  +  //-*)  c      '     a' +  b' 

245.  1.  1,  3,  5 ;  2.  $200,  300,  840 ;  3.  1,  2,  3,  4  cts. ; 
4.    $40,  30,  24,  2(>;  5.  $122|.i.  07 j.;.  liorsrs:  32,',. 
V2^  saddles ;  6.  65,  9,  49  years  ; 

7.  3  rt  —  2  r,  2  A  +  2  c  —  3  ^r,  3  a  -  2  A. 

249.  1.  11 .  1  ;  8,  3 ;  5,  5 :  2.  7  :  2.  0.  1  ;  2.  0  ;  3.  5.  1.1; 
2.  2,  1  ;  3,  1,  2;  1,  1,  3;  7.  7,  8,  0;  9.  Nu  values. 

250.  1.  55,  10;  15,  50;  2.  8.  2:   l."i.  1 :  22.  0.  etr. : 
7.  123,  224,  325,  etc.  729. 

255.    2.       .     "^   ,       ;  3.  ''i^l^:  r».  .\  -  •  ;  4.  45,  10; 

f  -\-  m  -{-  n  n  —  1 

lO4i,0O.U   300,-0;  5,  '"'-"'' -^  ^' :    3;    6.—^    :    T, ; 

//  -|-  1  ni  -\-  H 

b'r  —  be'        or'  —  u'l-      -,...,..     -..,  .,       „        nbc 

'■  laTT^b'  W^^.'  ii-'id.-'lr,:  8.  -^-^; 

9.  P^''  :  .; ;  10.  ^  (^^^  —  ^)     </  (//t  +  w)  .   -j^  , 
/'y  +  i^''  ~l~  fj>'  2  m  2  m 

11.^/-^;  6,  479. 

257.    1.  r/V^  121  /;*.-«.  10  w'A^V^j'  ./^r«.  -i^,  '*— —  , 

9.rV-'      4 

.,,„,..,„    fM)a*\  .      1        10  ,    ',.. 

\^b'^  )  2i«    225    ^ 

(^_  „)''  (  —  //)"  (—  r/,  !>"'.  ^,».  </V-"'  +  *.  if'^P-  ',  //- ""  '  "' ; 


15,v 


450  TEXT-BOOK   OF    ALGEBRA. 


2.  -  216  ^i«,  ^^^^^  b'Y,  a'',  -  8  a2i6'«, 

,i8a-3.^  /^y^.  3^  ^8yi-2^  ^|!^  729xV^  a^>^b'^c^%  9, 


729,  Y^oS  (—1)'"  t^^'"^^'V"\ 

260.  1.  i?2^2  —  2  ^^(/r  +  r2 ;  2.  8  w^^  —  12  /M'^yy  +  6  z/^/-  —  7/^ ; 

3.  ic«  +  12  xY  +  48  ic^y^  +  64  / ; 

4.  125  a^  —  75  6i2^,c  +  15  ab'e''  —  Pc^ ; 

5.  16  a'  +  32  a.^;r  +  24  a^^^r^  +  8  a^x^  +  a^x* ; 
Q,  I  -  S  b  -\-  24.1''  —  32  b^  +  16  b' ; 

7.  6t«  +  9  a«  +  36  (t'  +  84  a«  +  126  a'  +  126  a^  +  84  a^ 
+  36  (^2  +  9  «  +  1 ; 

8.  a*  -  f  <t^^>  +  I  a^b""  —  If  ab^  +  if  ^*; 

9.  625  —  2000  X  +  2400  ^r^  —  1280  x^  +  256  x' ; 

10.  8  ?-^  —  72  wr^  +  216  7n^'  -  216  m^ 

261.  3.  xY  +  2/'-'  +  ^'^''  +  2  ic V-  +  2  xi/z  +  2  icy;^'^ ; 

4.  1  +  3  a;2  +  3  £P^  +  x«  +  2  £C  +  4  ic^  +  2  ic^ ; 

5.  ^3  _^  3  ^^2^  _j_  3  ^^,2  _^  ^3  _  3^2^.  _  6  ^^^  —  3b''c  +  3  ac-^ 

_|_  3  jjc'^  _  c3;  6.  1  —  3  ri  +  5  a^  —  3  a^  -  a^; 

7.  1  _  6  :r  +  15  x^  —  20  £t;3  +  15a^*  _  6  ^^  +  ic« ; 

8.  8  (t^"'  —  12  a^^'b""  +  6  aH'^''  —  &«"  +  12  a^m^i^  _  12  «'"?/''c^ 
+  3  ^''"c^'  +  6  a"*  c^P  —  3  b^'c^-P  +  c^^ ; 

10.    a*  +  16  ^^^  +  c^  +  4  (2  a«^»  +  8  ab^  -  a^c  —  ae"  -  %  b\ 

—  2  bc^)  +  6  (4  a%^  +  a^c^  +  4  Pc'')  +  12  (2  a^^c^  -  2  a^^^c 

-  4  ab-'c). 

266.    1.  ^t  «^^^  i  ^^  ^*5  110  I'^al  root,  —  4,  -t-  3  x** ; 
2.  §,  i  -^,  i  ^ ,  -  '^^  imaginary,  i  —  ; 

3   Af^    _  6;  4.  —  7  a'^x^,  —  — ^,110  real  root,  no  real  root: 

6xY  ^y 

K  '^^^       0  .,8        2  1  9  A>2^4^ 

;r^  y  (vb^ 

1 

269.    2.  /«  +  3  o^^  3.  ^  (6^  +  %  n' ;  4.  a  --  ; 


ANS\VKi:t3.  451 

5.  ^'^  +  b'y\  g    ^,  _  ^  ^.  1 .  9.  ^^  1  _2; 

14.  „  _^  .:':i  -  —  4-  -^,  +,  etc. 

272.  1.  T)!,  217,  42.1,  20.82;  2.  6.42,  .Sl.()8,  4.1<)4,  .(>;i21 ; 

275.    l.'Ja  —  Sb;  2.  2  a  —  7  ./• :  3.  .r-  +  u;  +  1 ; 

4.  ,,-'  _  afj  -i-O'^i  6.  -  —  //  ;  9.  1  -''  —  —  —  .  etc 
//       '  ^        '-^ 

278.    1.  2.S,  2;U,  11.4,  5.51 ;  2.  .r)().S,  .2()().  i ;  3.  8.O20  -  ,* 
2.755  + .  1.710  —  ,  .585  -  ,  :5..S.S2  +  . 

281.    1.  'Jo  —  S  ./• ;  2.  1  —  ./  -f  ./ - :  3.  5  ./•  —  8  // ;  4.  2  —  X  ; 
7.  2  ./•  -  1 . 

288.    1.  11  -i-  7  =  18,  or  4:  2.  —  'rV..--^:  3.  15  „th^', 
4.  <;  </./•    "•  //*  4-  5  ^»c  +  ()  (^    -  ./•     ' ;  5.  0  : 

6.  6  rtJ-4  —  5  (./•  4-  t/)^  -j-  (>  (fr  —  //)*; 

7.  7  «x*'^  _  7  /,<•  —  2  ///*  +  6  r   "  4-  5  x>/ ; 

8.  8  c«(/«  +  10  c^rfJ  +  3  r*c/i  +  11  r8c/i ;  9.  -  327(y4. 

10.  216,  823543,  ^i^,  ^i,,  ^  ; 

11.  —  18  y^^,  18  ««//"  +  '•;  12.  2401  ;  13.  a«A  a^/j»h. 
14.  x^m^;  11^^;  15.  a^  -\-  a^  —  «-*  -f-  a-V  —  a^^'^  —  a^^; 
16.  «-'^»Wif/-?*;  19.  ///  —  y/ ;  21.  .r* ;  26.  <^*  —  3aV; 

-- 

26.  a"  -h  «T^-"'.-|-//-";  31.  u"'b-^    aibh>n     (-^  +  y)   '^ ; 

(x-  -  y)  , 
33.  j-iy- ',  <r  V  S  aiftVc   3 .  36.  1  -f  li  ^.     i  _  .s  ^r    «  +  4  j:    \ 
295.    1.  (2a6r)^  (3«</-^*,  (50)i;  2.  (9«2xy»)\  (10  a-//')*, 

298.    1.  12  (2)»,  10  (3)*,  4  (4)*,  9  (3)* ;  2.  3  a«  (j;)*,  6  a  (a)*, 
12  ab^  (3  «A)*,  5  h-.  (2  a)»,  4  aa;«  (5  ax)* ;  3.  4  aV,  (2  aZ^*^)*, 
—  3  x//  (4  x)i,  5  rf  (1  +  aZ»)i,  -  8  ay,  ar  (a  -f  bx% 
10  ./•'/ '."»  '/' >* ; 


452  TEXT-BOOK   OF    ALGEBRA. 


301.    1.  i  (:^)%  i  (10)^  1  (6)^  ^j  (21)^  1  (22)^  ^  (2) 


2   :^  r\>\i- 


1    u: 


^•'^^    ,,..     1 


2.  -(6  a^^c)  ^  J  (6  ao-)  ^  -^  (b)^ ,  ~  (3  a^»)S  I  (10)^ 


304.    1.  (_125a«^^)';  2.  (36)%  {J^a'b'')^ ,  [  ±]  ,  (49 //^.V-^)^ ; 


9 

3-  (i^j .  (0^  -  -')^)^  (216  .^«^-«)^  (^^^^  j  ; 

4.  (r/'^)«,  (a;6«)^,  (a^f,  (a% 

^07.    1.  (350)^  (1)^  (j^y,  (804)3,  (40)^  (i)^; 
2.  (5^»)^  (12a^ic2)^,  (ct^^/)^,  (o'«  +  2^;'"+2y«5 


310.    1.  (64)^,  (81)^  (6)«;  2.  (««)-,  (^')^'5  3.  (25)^  (64)^^; 

4.  (a')^,  {b)i',  5.  (6561) ^  (512)^  (15625)"; 

6.  (a'''y\  {a^y^,  (a^)^'. 

313.    1.  48  (5)^  107  (3)^ ;  2.  55  «//  (3  af  —  33  a//  (2  a^ ; 

3.  (20  a^  +  15  ^^2  +  4  c'^)  (7  ic)^ ;  4.  16  (11)^  20  (3)^  -  13  (2)^ ; 

5.  12i  (3)^;  6.  a  (3  a)^;  7.  3  (2)';  8.  9  (2  «)^. 
316.    1.  14  (6)-^,  12  (3)^,  10  a  (3)^  14  (9)^ ; 

2.  a^»"  (a&)^  ^  {2f,  30  o^t/^  ;  3.  ^  (6/i»)^  2  «^  140 ; 

4.  2  (5)^  6  (3)^  V-  (4)^;  5.  4  (80000) ^S  (648000)"; 

7.  (2  Z.'^c)*,  f  (12)^;  8.  (^^j   ,10  +  4  (35)^  +  30  (2)^ 
9.  5  (9  a«)^,  20  b  {aH>f,  (432  a'c')";  10.  6  a/^V  ; 

11.  (ic"/'-^?)"^,  y\h  •   12.  of-y  (a%^c^x'^i/*z'^y^ ; 

15.  3  (20)^  -  12  (3)^  -  (180)^  +  r2,  ab  +  "^ 
,    f  a    [    bd\      ,     , 


ANS\VKl:S. 


4:>:^, 


319.  1.  -^-^   ,  ^^^,  •       j^^      .-        ^,^^^    -     , 

G^l:>5)i  .^   (25000)*   (a^^)^   ,,. 

r>     '  '     10    '     ^ 

4.3[(0)»-f(fi>4-f(4)i];  5.  ^  -f  3M  -  (1 --a;V\^,._^,,.)i__,. 
9.  _  (1  7  +  L'  •  *j5;ii  +  S  .  'Al  -\-  ('»  •  L>«  +  4  •  ;^5  4.  i>  .  iji.sS). 

320.  1.  "..■'  (.^r.  -  10  (^-  I0)i ;  2.  ,^,  ^  (3)* ; 

3.  -  ^l^-  (2  :ry«)i,  -  ^  (^^)* '  ^-  "  ''  (^)*'  »>''  «>''  '  «  ' 

6.  125  (5) J,  (4)*,  (2  a)- ;  6.  3  (1  -h  2  x  +  «'),  26  -  15  (3)^ ; 

7.  x  -  2 a^V"'  +  y"*.  49  -  20  (6)*; 

8.  .r*  —  3  a-//  -|-  3  a-*//  —  //*,  ^    -       ^  ^— ^-  ; 

a' 

9.  (^  +  27  !/)x^  +  (9  ^  +  27  //)//*,  -  a*  +  (/..•)* ;   10.  {a')K  a^i 
11.  2  ^r-  (2  r)'**,  C,rV/)* ;   12.  2  «x'  (12)i ,  (7  «)*  . 

324.    1.  (l.')i  -  1,  5  -  2  (6)*,  3  (7)*  -  2  (6)*,  G^  -  2, 
2  (7)*  +  14* ;  2.  3  (7)*  +  2  (3)*,  2  -  ^  (3)*,  1  +  ^  (2)* ; 

3.  (a;  _  1)*  -l,x}-  (a-//)* ,  (1  +  a)^  -  (1  -  ^)* , 
ai  [(x  -  a)*-  «»),  (a-  +  //)*  +  -*; 

4.  1  +  3»,  1  -  5*,  2*  -  7^  4  -  7*,  1'^  +  3.  5  -  3* ; 

5.  1  +  'JK  1  +  5*,  i  (1  +  5i),  5()i  +  1«*. 
326.^    1.  14  (-  3)*  -  18  (-  2)i ;  2.  5  -  7  /; 

3.  3  (_  1)1,  (2  t)*  4-  (^  0* ;  4.  (-  2)*,  (/>»  -  .4)  /; 
6.-8  (6)*,  —  210 ;  6.  -  60  (42)*  i,  —  12 ;  7.  -  // '/.  ^/^ ; 
8.  5  (3)*,  2  -  i  (2)*i ;  9.  4  ;  -f-  5*,  ^l  CA)h . 

10.  ;,  —  2-2(^-3)*; 

11.  7()  _  ;;  (3()^*  —  [10  (5)*  +  21  (6)*]  /,  -  46  -  43  (-3)*. 

'  The  lettiT  i  is  ofti-n  written  for  y/—  I. 


454  TEXT-BOOK    OF    ALCKIUIA. 

329.    1.  14 ;  2.  3 ;  3.  4 ;  4.  10 ;  5.  7 ;  6.  144 ;  7.  .\  ;  8.  T) ; 

9.  1;   10.  ^il+i!;   11.  1;   12.  1;   13.  ^f^H^;   14.  V- 

a  2a  —  0 

332.    1.  i  2 :  2.  i  8 ;  3.  i  3 ;  4.  (62)^  i  5.  ^  4 ;  6.  ^  7 ; 
7.  -7^-T-;  8.  i  1;  9.  (ahy--,   10.  ^  M- 

338.  1.  G,  and  —  10 ;  2.  3,  and  -  25 ;  3.  9,  -  8  ;  4.  8,  2 ; 
5   1,  _  D ;  6.  6,  0 ;  7.  31,  —  11 ;  8.  15,  8 ;  9.  14,  -  2 ; 

10.  —  X   —  i;  11.  6,  —  5i;  12.  70,  50;  13.  2,  .48; 

14.  1  (3  i  (5)i)  ;  15.  A-  (-  19  i  58li)  ;   16.  i  (5  i  67^)  ; 

17.  ,V  (8.^)  i  451.96^)  ;  19.  2^,  -  7| ;  20.  |,  f^  ;   21.  ,%,  -  f. 

339.  1.  =k  2,  i  10 ;  2.  =t  7 ;  4.  7,  3 ;  5.  225,  121 ;  6.  27 ; 
7.  112  and  76 ;  8.  4  mi.  per  hour ;  9.  3,  4,  5,  and  —  1,  0,  1 ; 
10.  259,  481  yds. ;  11.  8;  12.  i  6;  13.  $7950;  14.  15, 17,  8  ft. ; 

15.  12 ;  16.  140,  120 ;  17.  160,  90 ;  18.  9  pence. 
343.    1.  4,  3,  and  5\,  If ;  2.  3,  5,  and  ^  72,  23| ; 

3.4,3;  -1,-20;  4.4,1;  7,10;  5.^12,3;  3,12; 

6.  5,6^',—  Jjf ,  —  71 ;  7.t  71,  13 ;  8.t  43,  —  51 ;  9.t  1,  1 ; 

10.  2,  3;   I,  4;  ll.t  12,  4;  12.  7,  4;    -  4,  -  7 ;  13.  5,  3. 

346.  l.t  il3,  il;  2.t  il,  il;  S-t  i2,  ^1; 

4.  ilOl,  i4;   i6,  i7. 

5.  i-a(i5^il),  ^./(i5^=pl); 

6.  i  (i(6  ahY^  U^fby^  y  it.  (i  (6  aiy  =p  (a/>)i) ; 

7.11,5;  6,10;  8.  9^,  3i;  - 2,  -  4;  9.  f  (1  ±  5^),  o^„(l  i 5^) ; 

10.  i2,i:l;   i4(7)^=FH7)^; 

11.  i  2,  i  1 ;  i  ^  ,  i  c>c  ,  (See  358,  2)  ;  12.t  i  |,  Azh', 
13.  ±7,i2;   ±3^^3(3)^ 

347.  1.  Az  36,  i  16  ;  2.  i  77,  :t  ^1 ; 

3.  1  (1  i  5^)  ;  i  (3  i  5^)  ;  4.  36 ;  5.  64,  36 ;  6.  $2025,  $900 ; 

1  This  problem  is  symmetrical  in  x  and  y;  i.e.,  tliey  can  cliange  places  without 
altering  the  equations  (2-30,  «)•  This  the  answer  indicates,  since  the  vahies  for 
X  and  y  will  satisfy  the  equation  when  interchanged.  Hereafter,  instead  of  writ- 
ing these  double  results,  the  answer  will  be  marked  with  a  dagger  to  show  that 
the  values  can  be  taken  the  other  way. 


ANSWERS.  455 

7.  i  10,  i  2  ;  i  6  (2)i,  i  4  (2)* ;  8.  $2,  $3  ;  60,  40  ; 
9.  4  ft.,  13  ft.  ;  10.  10  yds.,  2  yds.  ;  32  yds.,  1  yd. ; 
11.  S  Ins.,  0  lirs.  ;  12.  30,  38^  ;  28,  22^. 

351.    3.  3.  4  ;  4.  4,  T) ;  6.  —  T),  —  0 ;  6.  —  1,  -  12  ; 
7.  _l()i,,  -11^;  8.  +y,  -3;  9.  -10,  +30;  10.3,5; 
11.  21,-10;  12.  ^,  i. 

354.  1.  -r>./,  -18«;  2.0,  -3,  -^;  3.  ^iH«)*; 
4.  -  1,  i  i;  6.  i  2,  -  4  ;  7.  0,  i  4,  2;  8.  1,  2,  i- 

355.  2.  x«  -  7  a!  +  0  =  0  ;  3.  a;*  +  4  x»  -  7  j-2_  10  u-  =  0  ; 
4.  .1-   -  15  a-2  -h  60  X  —  46  =  0. 

357.    ,/  =  2,  a  minimum,  y  =  —  V,  »'*'  maximum. 

361.  5.1.-^^j-3)\-l.L^i^; 

6.  ./  2-*  (i  1  i  0  ;  7.  0  ;  8.  025  ;  9.  18.72  ;   10.  32 ;   11.  +  2 

362.  5.  2,  -  1 ;  6.  i  2,  i  1 ;  7.  ±  7,  i  5;  8.  i  1,  i  i; 
9.  [i  (7  i  349  *)]» ;  10.  9,  9* ;  11.  12*,  7* ;  12.  9,  6.76 ; 

13.  /-^:^(^'  +  4qc)*)y.  ^^  81,2401;  15.  343,  -J^^*^^; 

16.  256,  (-  24)' ;  17.  0,  4,  9 ;  18.  (§)* ,  (i)* ;  19.  5,  3,  4  i  10* ; 
20.  2,  1,  H3i(-31)*);  21.4,2,  H- 7  ±17*). 

363.  l.t  5,  2;  2.t  ±2(2)*,  ±5*;  S.f  ±  i,  ±  i ; 
4.t±3,  ±1;  6.  1944*,72»;  6.t7,4;  7.t5,3; 

ll.t  +1,-2;  12.t  7,  2 ;  13.t  27,  S ;  14.  81,  H ; 
16.  8,32;  2«,  4*;  16.t  ±4,  ±3;  17.  32,  i;  4,  -3;  18.  2,  1 ; 
19.2,2;  16,  i;  20.4,2;  3  ±  21*,  3  ^  21* ;  21.2,3;  Ti«,18; 
22.t  r,.  4 ;  8  ±  T^a  (-  2505)*,  8  =f  yij  (-  2505)*. 

364.  1.  8J,  2i,  5^  ft. ;'  2.  343,  64  cu.  ft. ;  3.  ±  3,  ±  1 ; 

4.  64,  512;  6.  ±  6,  ±  3;  6.  4,  2  ;  3  +  (-  19)*,  3-  (-  19)*. 

367.   8.  The  first;  9.  The  second. 


456  TEXT-BOOK    OF    ALGEBllA. 

368.    2.  .T  <  14 ;  3.  X  <  8,  y  >  3i  ;  4.  ^  >  a, 

and  {ah  —  a  +  h)  x  <  ah'^ ;  6.  0 ;  7.  S. 

379.    1.  I ;   2.  If  ;  2"*^,  l^t  ;   3.  3 ;    4.  6400 ;   5.  §  (3)^  ; 
6.  h  I,  e,  II ;  7.  2  :  1 ;  9.  -  i\  ;   12.  6,  8 ;  13.  | ; 
14.  1^-   15.  J,  6,  i  10;   16.  !!i2Lzi^; 

rt^  +  ^^  p  —  q 


m  —  n  —  i>  +  ^ 


380.  4.    (1.  56;   (2.  88f ;  (,3.  100;   (4.  ^; 

c 
(5.  25  x^  =  27  ?/=^. 

388.    2.  (1.  (1).  2,6599;  (2).  2.5378;  (3).  2.5647; 
(4).  2.6532;  (5).  1.6532;  (6).  1X021;  (7).  0.6021 ; 
(8).  0.4133;  (9).  2.5682 ;  (10).  3.6990;  (11).  5.8319; 
(12).  3.5623;  (3.  (1).  2.5073;  (2).  4.1512;  (3).  3.1512; 
(4).  0.4095;    (5).  3.3765;    (6).  1.3018  ;    (7).  2.5731 ; 
(8).  6.4632 ;  (9).  5.1232. 

390.    2.  .63329  ;  3.  .013015  ;  4.  2.8107  ;  5.  102.2 ; 
6.  17733 ;  7.  88.88 ;  8.  .0005395 ;  9.  .1099 ;  10.  .001051  • 
11.  .6955 ;  12.  .00111 ;  13.  .0003318. 

392.  3.  (2.  703 ;   (3.  2924 ;  (4.  28556 ;   (5.  2337200 
(2336544,  exactly). 

393.  3.  (2.  7;  (3.  5;  (4.  74.167;  (5.  14.342;  (6.  .004057: 
5.  (2.  .07094;  (3.  .001086;  (4.  523;  (5.  .004939. 

394.  3.  (2.  2401  +;  (3.  418333333;  (4.  2.051;  f5.  ^''>34.1 

(6.  22.8;   (7.  429.6;  (8.  .941. 

395.  2.  (3.  r:>.iS6(j  ;   (4.  .8806  ;   (5.  146.76  ;   (6.  5.4S75  : 
(1.  1.6155;   (8.  .70717;  (9.  4.957;  (10.  1.8217. 

398.    2.  (2.  6 ;  (3.  1.537 ;  (4.  .4581 ;  (5.  —  .2031 ; 
(6.  3.537;   (7.  1  ;   (8..-  .4319;   (9.  .9031;   (11.  3.011  : 
(12.  -t:  2.26;   (13.  4.336.  -  .336. 

402.    1.  (2.  —  -  a  ■   C3.  -  1.  ^/  i  S2  (Is  +  (a  —  i-  d)H^', 
n  '  ' 


ANSWERS. 


457 


(;j.  i  ,.  J-  a  4-  (,^  -  I)  ./;;   (4.  ^  ;.  yil  -  {n  -  l)d\; 

/i    I  — a.   /o  2(j;  — aw).    ,«       l^  —  a'^      .    ,.   2(w/--«) . 

^-  (1-  ;73T'  (^-  7(;r3T)'  (^-  25-/-^'  ^^'  7(73T)' 


4.(1.^^+1 


(2.  ^fr  ^i  ((2«  -  rfr  +  8rf*)4  -  2«  +  ^i;  (3.  -j^; 
(4.  ^  }2  /  +  ,/  ±  [(2  i  +  rf)'  -8rfs]»  i;  5.  (1.  l-{n-  1)  rf; 


/. 


403.    2.  ;U ;  3.  77,  779 ;  4.  -  71  :  5.  ^  (ii  +  5) ;  6.  142 ; 
7.  -  V»  -  1<>5 ;  8-  198  a  +  6.S  /y ;  9.  50,  5150;  10.  d  =  5  ; 
11.  2,  2.3  .  .  .  7.7,  8;  12.  43,-1;  13.  7,  6,  a'*;  14.  25'^ 
16.  106'\ 

407.    1.  (2.  (1)  ?L+i!lZllii;  (2)(^-^|"^^""N 

^•^-  '^  ^ ;:^'  (2)^^;^^^  C'^) ^-z- (—  i)-s-;  a.  (i) Q^; 
^-^ :^7 '  ^''  ^$^;  2.  (2.  (1)  ^-^-;-/;^'-  +  1 ; 

/  H  -  1   _  ^^  n  -  1 

(1>)  log-  (ff  +  (r  -  1)  s)  -  log,  ff  . 

log.  r 
(.S)  log-  /  "-  log.  (Ir  -  (r  -  1)  ^)  _^_  -^ . 

log.  r 

3.  ( 'J.  o  (s  —  ft)"-^  =  /(s  —/)'•- ^ : 


=  0. 


458  TEXT-BOOK   OF    ALGEBRA. 

410.  1.  6561,  9841 ;  2.  ,|^,  11H| ; 
3.  1,  ^1^,  4.  243  (6)^  364  (6^  +  2'^) ;  5.  ^\,  12^ ; 
6-  Uh  fill ;  7.  a  =  10,l==  1000000000 ;  8.  1953.1 ; 

9.  _  28,  14,  -  7,  I,  -  t,  I ;  10.  (a/j)h  •   12.  27 ;  13.  f  ;  14.  f|. 

418.  1.  12.68;  2.  $719;  3.  5%  ;  4.  lOi,  nearly; 
5.  3  yrs.  7  mos.  17  da. ;  6.  5^ ;  7.  2i,  nearly ; 
8.  A  little  more  than  21  years  from  1890 ;  9.  $8333 ; 

10.  Int.  $177.75 ;  11.  $5477.40 ;  12.  $1580.40  ;  13.  $1055  ; 
14.  $1276.76;  15.  $516.22;  16.  $1160;  17.  .0576;  18.  $1183. 


INDEX. 


Addition  — 

Of  Algebraic  Xumbers,  15-17, 

28-30 
Of   Algebraic    Quantities,    50, 

76,  a  ;  86-89 
Of  Fractions,  174 
Of  Quantities  with  Fractional 

Exponents,  287,  1  . 
Of  liiidicals,  311 
Of  luiaginaries.  325 
Of  Equations,  233 
Of  Inequations,  366,  5 

ADFKCTED    QlADKATK  ,  330 

Alokbka  — 

As  related  to  Arithmetic  and 

(;eonietr>',  2,  3,  4 
Definition,  7.     See  189,  r 
Annuity  Certain,  416 
Antecedent,  369 
Approximation,  396,  and   Re- 
mark, 417 
Arithmetical     Progression, 
399 

A  HR  AX<f  KM  HXT  — 

Of  INdynomial.  123 

Of  Letters  in  Term,  107,  a 
Associative  Law,  38,  2:  94 
Axiom,  207 

Bar.      See  Parenthesis. 
Base,  of  System  of  Logs.,  382 
Binomial  — 

Quantity,  70 

Suni,  323 


Binomial  — 
Square  of,  114 
Cube  of,  116,  5 
Any    Power,   Newton's   Theo- 
rem, 258 
Factoring  of,  134 

Bonds,  417 

Brace,  Brac  ket.     See  Paren- 
thesis. 
Brioos,  Logarithms,  385 
CuARAc  TKHisTic,  of  Logarithm, 

384,  388 
Coefficient  — 

Simple,  74 

Compound,  74,  75 

Of  l^dical,  297,  2 
Commi'tative  Law  — 

In  Addition,  28 

In  Multiplication,  38 

Comparison,  Elimination,  231 
Complete  Quadratk  .  330 
Complex  — 
Fraction,  155,  1 
Number,  325,  Remark. 

COMPOINI)  — 

Terms,  68,  h  ;  96-98 
Composite  Numher,  132,  4 
Conditional,  Equation,   188- 

189 
C0N.J10ATE,  Si  RD,  355,  (/ 
Consequent,  in  Proportion,  369 
Corollary,  203 


>  Can  be  um.i  i.-  » 
the  i>tu<if  nt'x  proirrc! 


,  i«\vx  l.y  i^uoring  hII  nf»T«-n«M*'«  in  ;i(l\  uiict-  of 


459 


460 


INDEX. 


Cube,   and   Cube  Root,   57,    h  ; 

60,  a 
Cyclic  Order,  177,  Ex.  19 
Degree  — 
Of  a  Term,  77 
Of  an  Equation,  195 
Demonstration,  202 
Development,  258 
Difference,  17 
DioPHANTiNE  Equations,  246: 

Chapter  XVI.,  339,  3(5 
Discussion  of  Problems,  356, 

358,  359 
Dissection  of  a  Fraction,  163 
Distributive  Laav,  109,  5 
Divisibility    of    Quantities, 

130 
Division,  118.      See  Multiplica- 
tion. 
Duplicate  Ratio,  372,  a 
Elimination,  226,  226,  a  ;  227 
By  Substitution,  228 
By  Comparison,  231 
By  Addition  and  Subtraction, 

233 
By  Undetermined  Multipliers, 

236 
By  One  or  More  Derived  Equa- 
tions, 237 
By  finding  h.  c.  f.,  238 
In  Quadratics,  340 
When  one  Equation  is  Quad- 
ratic, 341 
Special  Methods,  345 
In   Equations  of   Higher  De- 
grees, 363 
Equations  — 

Definitions,  55,  185,  189,  e 
Identical,  187 
Conditional,  188,  225,  a 
Simple,  196, 
Quadratic,  330 
Higher,  353,  361 
Exponential,  381,  398 
Type-forms,     208,     331,    333, 

344,  a  ,-  362,  381 
Independent,  246 


Equations  — 
Redundant  — 

Compatible,  251,  1 

Incompatible,  251,  2 

Indeterminate,  225,  h  ;  246 
Radical,  327 
Properties  of,  348-354 
Construction  of,  355, 
Solutions  of,  215,  239,  etc. 
Error,    of    Log.    Tables,    396, 

415,  Remark. 
Evaluation    of    Alg.    p]x"s. 

Sec.  III.",  Chap.  IV. 
Evolution,  262 
Expansion,    same   as    Develop- 
ment. 
Explicit  Function,  357,  a 
Exponent,  56,  57 

In  Multiplication,  106,  3 ;  117, 1 
In  Division,  119,  129 
Zero  and  Negative,  128 
Fractional,  282,  et  seq. 
Expression,  Algebraic,  48 
Simple,  68,  a 
Compound,  68,  h 
Quantity,  95,  95,  a 
Extremes,  373,  b 
Factors,  132 

In  Multiplication,  34 
Order  of,  38 

Zero,  111,  ^/ ;  128,  Note. 
Finding,  132,  et  seq. 
Prime,  132,  3 
Common,  Chapter  XI. 
Figures,  7,  a 

Formulae,  113,  a.  See  114, 
130,  254,  255,  258;  337, 
4.  These  are  only  a  few  of 
the  most  important  for- 
mulae. 
Fraction  — 

An  Indicated  Division,  154,  h 
As  Exponent,  282 
Function,  Def.,  221.     See  222, 
(in (I  222,  (( 


(iEOM  ETHICAL 

404 


Pkoghession, 


1  M  » 1 .  X  . 


4H1 


IIioHKST  Common  Factor,  139. 

2,  et  se<i. 
IIoM<HiKXKors  — 

Quantities,  78,  a 

E<juations,  345,  '.\ 
lIvi'oTiiKsis,  205 
Idkxtical  E<iUAT10XS,  187 
Imaoinakiks  — 

Ori^'in,  45,  1.  (2 

SipiiticaiuM',  265. 325,  Keniaik. 

FiuuianuMital  OiMTations  with. 
325 

Imaginaries  in  Pairs,  355,  a 
Implicit  Function,  357,  a 

iNCOMMKNSrilABLK  Ql  ANTI- 

TIKS,  290,  a 

INCOMPLKTE  QlADHATK  ,  330 

Inoktkhminatk  equations, 
Chapter  XVI.,  339,  80 

Indkx  — 
Of  Power.  57,  '/ 
Of  Root,  59 

iNKt^l'AMTY,  365.  H  Kftf. 
INFINITK     (^lAXTITY,    62,     358, 
386 

Infinitesimal,  62,  358 

Inhpection,   Solution   by,    115. 
etc. 

I  XT EK  EST.  412 

Simple.  216.  Ex.  47;  also  413 
Annual.  414 
Compound,  415 

Interpol atiox,  388 

Invohtion,  256 

Irrationals,  290,  290,  a 

Like  — 

Sij;ns,  26,  76,  h 
Tenns  (oftener  called  ximilar), 
76,  311,  a 

Limits,  368 

Literal  — 
(Quantities,  75 
E<iuations,  192 

Lo(iARiTliMS,  Chaptrr  XX  \i. 


Lowest    Common     Miltiple, 

Chapter  XII. 
Maxtissa  of  Lo(;ai{!TIIm,  383, 

386 
Maximum  Value  of  function, 

357 
Meax  — 
Arithmetical,  402 
(Jeometrical.  378,  407 
Memhek  — 
Of  E(|uation.  190 
( )f  Proportion  =^  Term. 
Ml XI MUM,  367 
Minus,  26.     See  Neyative. 
Multiple,  149,  1 
Multiplication,  104.    Sec  A^l- 
ilition    for     Stibjerts    ami 
Ajtproximufe  lieferenres. 
By  Logarithms,  392 
Napieh,  Inventor  of  Logarithms, 

386 
Nkoative  — 
Quantities,  7 
Series,  11.     Mark,  20 
Name,  25 
Addition,  etc.,  of,  28,  31.  34, 

35,43 
Eximnent,  128 
Solutions,  253 

Plus  and   Minus,  45,  h;   264. 
344,  Remark. 
Newton's  Theore.m,  258 
Normal  — 

Process  of  Solution  of  Simple 

P^quations,  215,  1 
Fonn  of  Quadratic,  349 
Nought.     See  Zero. 
Numerical  — 

Cwfticients,  73,  106,  2 
Value,  81 
Operation,    24,    63,    325,    II.- 

mark. 
Parenthesis.    54.   65.  Chapter 

VIl. 
Plus,    26.       -         \   jatire  for 
Topicfi. 


462 


INDEX. 


Polynomial,  72 

Square  of,  116,  2 

Addition  of,  87,  etc. 

Factoring,  136,  137 
Positive.     See  Negative. 
Power,  Chapter  XVII. 

Definition,  57,  128,  a 

Of  Algebraic  Number,  42 

Of  Algebraic  Quantity,  117 

Of  Unity,  Remark,  117 

Of  Radicals,  320 

Of  Imaginaries,  325 

Of  Terms  of  Proportion,  376,  4 

By  Logarithms,  394 
Prime  — 

Quantity,  132,  3 

Factors,  140 
Problem, 200 

PROGRE8SIOXS,  399,  404 

Property  — 

Of  Algebraic  Numbers,  34,  a 
Of  Identical  Equations,  187,  a 
Of  Quadratic  Equations,  349, 
350 
Proportion,  373 

In  Equations,  223 
Pure  Quadratic,  330 
Quadratic,  330 
quadrinomial,  136 
Quantity,  1,  a ;  95 
Q.  E.  D.     Quod  erat  demonstran- 
dwii^    "Which  was  to  be 
proved." 
Radicals,  Chapter  XX. 
Ratio,  369 
Rational.  290.  a 
Ratioxalizatio.n    of    1)p:.\om- 
INATORS,  317 

Real  Qi  antity,  9,  45,  1  (2 

1»E(  II>ROCAL.   128,   h 

Reduction,  158 

Of  Radicals,  292 

Rationalization,  317 

Of  Repeating  Decimals,  409 
Root  — 

Of  Quantity,  60.      Sec   Poircr. 


Root  — 

Of  an  Equation,  193 
Satisfy  an  Equation,  189,  a 
Scholium,  204 
Signs,  20,  49,  45,  b ;  64,  358 
Similar  — 

Terms,  76,  287,  1 

Radicals,  311 

SlMPLPJ  — 

Term,  68,  a 
Equation,  196 

Simultaneous  Equations.  See 
Elimination. 

Sinking  Fund,  416,  1  (2 

Solution  of  an  Equation,  193 

Square,  57,  h ;  -Root,  60,  a 

Substitution  — 
Elimination  by,  228 
Principle  of,  113.  This  prin- 
ciple is  much  used  in  al- 
gebra. It  constitutes  one 
of  the  advantages  of  the 
literal  notation.  Refer- 
ences, 95,  112,  113;  208, 
254,  a;  258,  259,  3;  344, 
4;  362,  Exs.  1,  3,  11,  19, 
23,  30,  39;  363,  Exs.  13, 
16,  29;  397;  398,  13 

Subtraction,    90.      See   Befer- 
ences  for  Addition. 

Surd,  290 

Symbol,  61.  Classification,  62-67 

Symmetry,  Def.,  230,  a 

Examples,  363,  1-4,  (>,  7,  10, 
11,   12,  13,   1(),  20,  22,  2.5, 

28 

System  — 

Of  Equations,  225 
Of  Logarithms,  383 

Terms,  68,  69,  Sec.  lY.,  Chapter 
ill. 
Of  a  Fraction,  154,  a 

Theorem,  201 

See  114,  116,  128,  130,  258, 
322,  366,  367,  375-377, 
397 


INDEX. 


463 


In  E(i nations,  210 
In  Inequalities,  366,  1 

TniNOMIAL,  71 

Square  of,  116,  1 

Cube  of,  261,  1 

Square,  114 

Factoring  of,  135 
Tripi.u  ATK  Hath),  372,  a 
Type-Fokms.     See  Equation. 

UXDETKKMINKI)     MVI-THM-IKKS, 

236 

Unity  — 

As  Coefficient,  75,  h 
As  Exponent.  56.  o 


L'mtv  — 
As    Denominator  of    Integer, 
169 
Unlike  Signs.    -See  Like  Siijns. 
Value,  Num.  81 ;  Max.  357 
Variation,  380 
Verification,  189,  a 
Vinculum.     <See  Parentheidtt. 

Zero,  19,  62,  a 
As  Coefficient,  111,  a 
As  Exponent,  128 
Log.  of  Equals  —  x  ,  386 
As  Log.,  383,  a 
As  Divisor,  358.  3 
As  Dividend.  358,  1 


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